Physics Book - User contributions [en]

Polarization of a conductor

2016-11-25T15:38:47Z

Mtikhonovsky3:

Edited by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

(''Matter and Interactions'')

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

(''Matter and Interactions'')

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

1) Mobile Charges

2) Polarization

3) Equilibrium

4) Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:38:27Z

Mtikhonovsky3: /* 3. Difficult */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

(''Matter and Interactions'')

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

(''Matter and Interactions'')

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

1) Mobile Charges

2) Polarization

3) Equilibrium

4) Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:37:49Z

Mtikhonovsky3: /* Ionic Solutions */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

(''Matter and Interactions'')

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

(''Matter and Interactions'')

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

1. Mobile Charges

2. Polarization

3. Equilibrium

4. Excess Charge

====Solution====

1. Conductors have mobile charges while insulators do not

2. Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3. Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4. Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:37:19Z

Mtikhonovsky3: /* Mobile Electron Sea */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

(''Matter and Interactions'')

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

1. Mobile Charges

2. Polarization

3. Equilibrium

4. Excess Charge

====Solution====

1. Conductors have mobile charges while insulators do not

2. Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3. Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4. Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:36:45Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

1. Mobile Charges

2. Polarization

3. Equilibrium

4. Excess Charge

====Solution====

1. Conductors have mobile charges while insulators do not

2. Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3. Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4. Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:36:18Z

Mtikhonovsky3: /* 3. Difficult */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

1. Mobile Charges

2. Polarization

3. Equilibrium

4. Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:35:33Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(''Matter and Interactions'')

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:33:51Z

Mtikhonovsky3: /* References */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. ''Matter and Interactions''. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-25T15:33:24Z

Mtikhonovsky3: /* References */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2015. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:54:51Z

Mtikhonovsky3: /* References */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:52:35Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, <math>{Δt}</math> is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the <math>{Δt}</math> in the equation. <math>{Δt}</math> refers to the AVERAGE time between collisions. In this case, the <math>{Δt}</math> would therefore be 7.5 seconds.

Plugging these numbers in, you get: <math>{1.3}{*}{10^{13}}{\frac{m}{s}}</math>. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:51:07Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, <math>{E_{net}} = {10}{\frac{N}{C}}</math>, <math>{m_{e}} = {9.1}{*}{10^{-31}}</math>.

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:49:35Z

Mtikhonovsky3: /* 2. Middling */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

All of the quantities are known. <math>{e} = {1.6}{*}{10^{-19}}{C}</math>, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:47:17Z

Mtikhonovsky3: /* Examples */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of <math>{10}{\frac{N}{C}}</math> is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:46:13Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = <math>{0.333}{\frac{N}{C}}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:45:45Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045}</math> = 0.333 <math>\frac{N}{C}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:45:21Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}{u}}</math> = <math>\frac{0.0015}{0.0045E</math> = 0.333 <math>\frac{N}{C}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:44:57Z

Mtikhonovsky3: /* 1. Simple */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s)}{(N/C)}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045E</math> = 0.333 <math>\frac{N}{C}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s)}{(N/C)}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:44:18Z

Mtikhonovsky3: /* Examples */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===1. Simple===

The mobility of electrons in copper is <math>{0.0045}{\frac{(m/s}{(N/C}}</math>. How much net electric field would be needed in order to give the electrons in copper a drift speed of <math>{0.0015}{\frac{m}{s}}</math>?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045E</math> = 0.333 <math>\frac{N}{C}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{(m/s}{(N/C}</math>.

===2. Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===3. Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:41:58Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{0.0015}{0.0045E</math> = 0.333 <math>\frac{N}{C}</math>.

This also makes sense from a units perspective, as our solution is in <math>\frac{N}{C}</math>, the speed is <math>\frac{m}{s}</math>, and the mobility is given in <math>\frac{\frac{m}{s}}{\frac{N}{C}}}</math>.

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:39:42Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = <math>\frac{1.5E-3}{4.5E-3}</math> = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:38:44Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac{\bar{v}}{u}}</math> = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:38:20Z

Mtikhonovsky3: /* Solution */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

:: <math>\bar{v}={u}{E_{net}} </math>

Thus, <math>{E_{net}} = {\frac\bar{v}{u}}</math> = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:36:33Z

Mtikhonovsky3: /* Mobile Electron Sea */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>

where <math>\bar{v}</math> is the drift speed, <math>e</math> is the charge of the electron, <math>{E_{net}}</math> is the net applied electric field, <math>{Δt}</math> is the average time between collisions, and <math>{m_{e}}</math> is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where <math>{Δp} = {F_{net}}{Δt}</math> and <math>{F} = {e}{E_{net}}</math>.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:33:16Z

Mtikhonovsky3: /* Mobile Electron Sea */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}}} </math>
'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:32:55Z

Mtikhonovsky3: /* Mobile Electron Sea */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by <math>\bar{v}</math>, the drift speed, as mentioned earlier. For metals this equation is:

:: <math>\bar{v}={\frac{{e}{E_{net}}{Δt}}{m_{e}} </math>
'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:30:10Z

Mtikhonovsky3: /* Ionic Solutions */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, which in turn creates its own electric field! This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:29:29Z

Mtikhonovsky3: /* Charge Motion in Metals */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Metals===
====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:28:43Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:28:05Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge <math>{\frac{(m/s)}{(N/C)}</math>, and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:26:42Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge ((m/s)/(N/C)), and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, <math>u</math>, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the charge will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:25:40Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bar{v}={u}{E_{net}} </math>

where <math>\bar{v}</math> is the drift speed, <math>u</math> is the mobility of the charge ((m/s)/(N/C)), and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:24:40Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\line{v}={u}{E_{net}} </math>

where <math>v</math> is the drift speed, <math>u</math> is the mobility of the charge ((m/s)/(N/C)), and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:23:56Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\bf{v}={u}{E_{net}} </math>

where <math>v</math> is the drift speed, <math>u</math> is the mobility of the charge ((m/s)/(N/C)), and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:21:25Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\dash{v}={u}{E_{net}} </math>

where <math>v</math> is the drift speed, <math>u</math> is the mobility of the charge ((m/s)/(N/C)), and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:14:49Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>\vec{v}=mu{E_{net}} </math>

where v is the drift speed, <math>u</math> is the mobility of the charge ((m/s)/(N/C)), and <math>E_{net}</math> is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:12:52Z

Mtikhonovsky3: /* A Mathematical Model: Drift Speed */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

:: <math>{v}=mu{E_{net}} </math>

where v is the drift speed, u is the mobility of the charge ((m/s)/(N/C)), and E_{net} is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T21:08:27Z

Mtikhonovsky3: /* Ionic Solutions */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. The net electric field in the region is the superposition of the applied (external) field and the electric field created by the relocated charges in the material. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions are constantly moving in the solution, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

'''v = u*Enet'''
:: <math>{v}=mu{E_{net}} </math>

where v is the drift speed, u is the mobility of the charge ((m/s)/(N/C)), and Enet is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T20:25:20Z

Mtikhonovsky3: /* The Main Idea */

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material. Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous salt solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles move in the direction of the force applied by the electric field. However, the interesting concept here is that the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions move around the solution at all times since it is a conductor, there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

'''v = u*Enet'''

where v is the drift speed, u is the mobility of the charge ((m/s)/(N/C)), and Enet is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-24T20:24:04Z

Mtikhonovsky3:

Claimed by Margaret Tikhonovsky Fall 2016
==The Main Idea==

A conductor contains mobile charged particles that move freely through a material.Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed in: ionic solutions and metals.

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles (as you'd imagine) move in the direction of the force applied by the electric field. However, the interesting concept here is that the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions move around the solution at all times (it is a conductor after all), there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

'''v = u*Enet'''

where v is the drift speed, u is the mobility of the charge ((m/s)/(N/C)), and Enet is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Polarization of a conductor

2016-11-02T04:34:51Z

Mtikhonovsky3:

Claimed by Margaret Tikhonovsky 2016
==The Main Idea==

Based on the definition of a conductor, it is easily assumed that stronger conductors can have charged particles moving more freely within it and in larger distances. There are two main situations where this can be observed. The first is in ionic solutions:

===Ionic Solutions===

Ionic solutions, such as KCl or NaCl or NaI (solutions in which the ions dissociate), have individual ions of the dissociated particles. For example, a solution of NaI will have Na+, I-, and because it is an aqueous solution, some H+ and OH- as well. When an electric field is applied to this solution, the particles (as you'd imagine) move in the direction of the force applied by the electric field. However, the interesting concept here is that the charged particles move in a direction and accumulate, therefore creating a charge gradient, '''which in turn creates its own electric field!''' This can be shown in Figure 1, taken from the textbook. While ions are constantly moving, this is an accurate snapshot of what would be expected in the event of applied field/force. Furthermore, while ions move around the solution at all times (it is a conductor after all), there is a measurable excess of ions on the edges that generates the field.

'''The net electric field is the superposition of the applied field and the field generated by the charge gradient and excess ion concentrations on the edges.'''

[[File:Shah1.jpg]]

Figure taken from Matter and Interactions 4th Edition

====A Mathematical Model: Drift Speed====

In the phenomenon described above, when the external field is applied on an ionic solution, the Na+ and I- ions will move and bounce around a good bit, but due to collisions, they do not maintain a specific speed or trajectory. This is in spite of whether or not the force experienced is constant. In order to keep the ions moving at constant speed, also known as the drift speed, a constant electric field must be applied. This is modeled mathematically using the following equation:

'''v = u*Enet'''

where v is the drift speed, u is the mobility of the charge ((m/s)/(N/C)), and Enet is the net electric field applied to the ionic solution. The proportionality constant, u, is determined for the ions based on the solution and will be usually given or easy to derive in all practical problem sets using this concept. This is a linear relationship overall, meaning that in the event of no electric field, the ions will stop moving.

====Polarization Process in Ionic Solution====

While polarization is a rapid process once initiated, it is not an on-off binary process. The instant that an electric field is applied, the drift speed is nonzero and the particles start tending towards the direction of the electric field. As ions pile up on the sides of the solution, the electric field inside becomes weaker and the system approaches an equilibrium at a microscopic level. At this equilibrium, drift speed is 0 and there is no NET MOTION of mobile charges in solution

===Charge Motion in Metals===

The second example to look at for conductors is in metals.

====Mobile Electron Sea====

Metals are an interesting idea structurally. Atoms of a metal are arranged in a rigid, ordered structure, sometimes known as a lattice structure (similar to crystal). Due to this, there is a interesting pattern of behavior. The inner electrons of each of these ordered atoms are attracted to the positively charged nucleus. Some of the outer electrons engage in solid atom interactions (ball and spring model), but other outer electrons participate in a pool of electrons that is allowed to move freely across the solid, in an "electron sea". While they can move across the solid, they are very hard to extract and there are obviously some limitations to its direction of movement (different from liquid ionic solutions), but the mobile electron sea makes metals great conductor candidates (and thus are used in so many practical applications, including wires, microchips, cars, etc.)

There is a caveat to this, however. You may ask, well won't the electrons repel each other and this not really allow for movement? Yes, but due to the presence of the positive cores, this interaction is neutralized. There is, therefore, no net interaction between the electrons inside a metal. Thus, the sea electrons move freely independent of electron field, positive cores, and appear not to interact with each other. The model of electron motion is defined by the Drude model, the steps of which are shown here:

1) When an electric field is applied in a metal, the electrons get excited and accelerate.

2) The electrons lose their energy when they collide with the lattices of the positive atomic cores

3) After the collision, the electrons again get accelerated and the entire process repeats itself.

[[File:Shah2.JPG]]

Figure taken from Matter and Interactions 4th Edition

The average speed, then is defined by v, the drift speed, as mentioned earlier. For metals this equation is:

'''v = (e*Enet*deltaT)/(m)'''

where v is the drift speed, e is the charge of the electron, Enet is the net applied electric field, deltaT is the average time between collisions, and m is the mass of the electron. The average time between collisions is used because the time between each collision is uncontrollable and inconsistent. This equation is derived from the momentum equation where p = F*deltaT and F = Enet*e.

This YouTube video does an excellent job of explaining the concept in further detail:

https://www.youtube.com/watch?v=HKgOpmX-OFI

==Examples==

===Simple===

The mobility of electrons in copper is 4.5E-3 (m/s)/(N/C). How much net electric field would be needed in order to give the electrons in copper a drift speed of 1.5E-3 m/s?

====Solution====

This is a simple problem using the formula given for drift speed

'''v = u*Enet'''

Thus, Enet = v/u = 1.5E-3/4.5E-3 = 0.333 N/C.

This also makes sense from a units perspective, as our solution is in N/C, the speed is m/s, and the mobility is given in (m/s)/(N/C).

===Middling===

In a simple metal lattice structure, an electric field of 10 N/C is applied and 15 collisions are observed. The time between collisions increases after each collision, starting with 1 second, from collision 1 to collision 2. After 15 collisions, what is the drift speed of an electron in this metal lattice structure?

====Solution====

This is a slightly more complex problem that requires a bit of analytical thinking and application of a formula. The formula:

'''v = (e*Enet*deltaT)/(m)'''

All of the quantities are known. e = 1.6E-19 C, Enet = 10 N/C, m = 9.1E-31

However, deltaT is not immediately discernible. The question asks for the drift speed after 15 collisions. It might be some people's thought to just use the time between the 14th and 15th collision as the deltaT in the equation. DeltaT refers to the AVERAGE time between collisions. In this case, the deltaT would therefore be 7.5 seconds.

Plugging these numbers in, you get: 1.3E13 m/s. This is extremely fast. As you can see, this speaks to how quickly electrons move through substances and how quickly polarization occurs, both in metals and in ionic solutions.

===Difficult===

Summarize the difference between conductors and insulators in the following 4 categories:

Mobile Charges
Polarization
Equilibrium
Excess Charge

====Solution====

1) Conductors have mobile charges while insulators do not

2) Sea or mass of mobile electrons move when electric field is applied to a conductor. When an electric field is applied to an insulator, INDIVIDUAL atoms polarize.

3) Net electric field is 0 in a conductor at equilibrium whereas there is a nonzero electric field in an insulator at equilibrium.

4) Excess charge spreads over the surface of a conductor but clumps in patches in an insulator.

(Adapted from textbook)

==Real World Application==

While this isn't exactly related to conductors (the exact opposite, actually), this is a very important concept in polarization. As mentioned earlier, when a conductor is polarized, electron seas move rapidly and cause polarization. When an insulator has an applied electric field, the individual atoms or molecules become polarized. An interesting aspect of this is the concept of dielectric. A dielectric is a material or substance that is usually an insulator, but when an electric field is applied, the electrons shift slightly from their usual equilibrium creating a positive and negative space and thus, polarization. Usually, a dielectric is a material with EXTREMELY HIGH polarizability, which is extremely useful in the real world. The concept was developed by William Wheewell during a discussion with Michael Faraday. Diaelectrics are commonly used in developing capacitors.

== See also ==

===External links===
YouTube Video: [https://www.youtube.com/watch?v=HKgOpmX-OFI]

IEEE Report: [http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6507323]

==References==

Matter and Interactions 4th Edition by Chabay and Sherwood

[[Category:Textbook]]

Head-on Collision of Equal Masses

2015-12-06T02:47:16Z

Mtikhonovsky3: /* History */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

https://trinket.io/embed/glowscript/37540ee8e0

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

== See also ==
http://www.real-world-physics-problems.com/physics-of-billiards.html

http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/math.html
===Further reading===
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

http://www.physicsclassroom.com/class/momentum/Lesson-2/The-Law-of-Action-Reaction-(Revisited)

==References==
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.
This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-06T02:46:41Z

Mtikhonovsky3: /* External links */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

https://trinket.io/embed/glowscript/37540ee8e0

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==
http://www.real-world-physics-problems.com/physics-of-billiards.html

http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/math.html
===Further reading===
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

http://www.physicsclassroom.com/class/momentum/Lesson-2/The-Law-of-Action-Reaction-(Revisited)

==References==
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.
This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-06T02:39:36Z

Mtikhonovsky3: /* A Computational Model */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

https://trinket.io/embed/glowscript/37540ee8e0

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==
http://www.real-world-physics-problems.com/physics-of-billiards.html

http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/math.html
===Further reading===
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.
This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-06T02:39:05Z

Mtikhonovsky3: /* A Computational Model */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]
https://trinket.io/embed/glowscript/37540ee8e0

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==
http://www.real-world-physics-problems.com/physics-of-billiards.html

http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/math.html
===Further reading===
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.
This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-06T02:29:15Z

Mtikhonovsky3: /* References */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==
http://www.real-world-physics-problems.com/physics-of-billiards.html

http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/math.html
===Further reading===
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.
This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-06T02:28:46Z

Mtikhonovsky3: /* See also */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==
http://www.real-world-physics-problems.com/physics-of-billiards.html

http://archive.ncsa.illinois.edu/Classes/MATH198/townsend/math.html
===Further reading===
Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-06T00:08:19Z

Mtikhonovsky3: /* Connectedness */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
This as a real world application of billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards the other one which is stationary. When collision occurs, the ball thta is hit will move at 90 angle from the direction the stricking ball comes. Keeping this mind and with a little practice one's level of playing billiard should increase.

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-05T23:58:53Z

Mtikhonovsky3: /* External links */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html

http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Head-on Collision of Equal Masses

2015-12-05T23:58:34Z

Mtikhonovsky3: /* External links */

Work in progress by mtikhonovsky3
==Main Idea==

A collision is a brief interaction between large forces. A head-on collision can be between two carts rolling or sliding on a track with low fricton or billiard balls, hockey pucks, or vehicles hitting each other head-on.

In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final ''x'' momentum ''p1xf'' + ''p2xf'' must equal the initial total ''x'' momentum ''p1xi''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K1i'', and the internal energies of both carts. After the collision there is internal energy of both carts and kinetic energy of both carts, ''K1f'' + ''K2f''.

===A Mathematical Model===
Head-on Collisions of Equal Masses can be based off the Fundamental Principle of Momentum:
::<math>{\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt}</math>
where '''p''' is the momentum of the system, '''F''' is the net force from the surroundings, '''Δt''' is the change in time for the process.
Energy Principle:
::<math>{E}={Q}+{W}</math> where '''E''' is the total energy, '''Q''' is the heat given off, and '''W''' is the work done.
If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

Based off of the Momentum Principle and the Energy Principle, we will explore Head-on Collisions of Equal Masses in two different scenarios: elastic and maximally inelastic (objects become stuck together).

'''1. Elastic Head-on Collisions of Equal Masses'''

Momentum Principle:
::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}+{0}</math>

Energy Principle:
:: <math>{E_f}={E_i}+{W}+{Q} </math>
:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
Since ''K=p2/2m'', we can combine the momentum and energy equations:
::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>
:: <math>{p^2_{1xf}}+{p^2_{2xf}}={p^2_{1xf}}+{2p_{1xf}p_{2xf}}+{p^2_{2xf}} </math>
:: <math>{2p_{1xf}p_{2xf}}={0} </math>

There are two possible solutions: ''p1xf =0'' or ''p2xf =0''.
If ''p1xf =0'', then object 1 came to a full stop. Based on the momentum equation ''p1xf + p2xf = p2xf = p1xi'', then object 2 now has the same momentum that object 1 used to have. There is a complete transfer of momentum from object 1 to object 2, and so, there is also a complete transfer of kinetic energy from object 1 to object 2.
If ''p2xf =0'', then ''p1xf + p2xf = p1xf = p1xi'', and so object 1 keeps going, missing object 2. This won't happen if the carts are on the same track. It is not possible for both final momenta to be zero, since the total final momentum of the system must equal the nonzero total initial momentum of the system.

'''2. Maximally Inelast Collision of Equal Masses'''

Momentum Principle:
::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>
::<math>{2p_{1xf}}={p_{1xi}}</math>
::<math>{p_{1xf}}={\frac{1}{2}}{p_{1xi}}</math>

The final speed of the stuck-together carts is half the initial speed:
::<math>{v_{f}}={\frac{1}{2}}{v_{i}}</math>

Final translational kinetic energy:
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}{v^2_{f}})</math>
::<math>({K_{1f}}+{K_{2f}})={2}({\frac{1}{2}}{m}({\frac{1}{2}}{v_{i}})^2)={\frac{1}{4}}{m}{v^2_{i}}</math>
::<math>({K_{1f}}+{K_{2f}})={\frac{K_{1i}}{2}}</math>

Energy Principle:
::<math>{K_{1f}}+{K_{2f}}+{E_{int,f}}={K_{1i}}+{E_{int,i}} </math>
::<math>{E_{int,f}}-{E_{int,i}}={K_{1i}}-({K_{1f}}+{K_{2f}})</math>
::<math>{ΔE_{int}}={K_{1i}}-{\frac{K_{1i}}{2}}</math>
::<math>{ΔE_{int}}={\frac{K_{1i}}{2}}</math>

The Momentum Principle is still valid even though the collision is inelastic, and fundamental principles apply in all situations. The final kinetic energy of the system is only half of the original kinetic energy, which mean that the other half of the original kinetic energy has been dissipated into increased internal energy of the two carts.

From both examples, we know know:
1. If the collision is elastic, object 1 stops and object 2 moves with the speed object 1 used to have.
2. If the collision is maximally inelastic, the carts stick together and move with half the original speed. Half of the original kinetic energy is dissipated into increased internal energy.

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

A 8 kg mass traveling at speed 18 m/s strikes a stationary 8 kg mass head-on, and the two masses stick together. What are the final speeds?

::<math> {m}{v_{1i}}+{m}{v_{2i}}={m}{v_{f}}+{m}{v_{f}}</math>
::<math> {m}{v_{1i}}+{m}({0})={2m}{v_{f}}</math>
::<math> {8 kg}({18m/s})={2}({8kg}){v_{f}}</math>
::<math> {144kg}{m/s}={16 kg}{v_{f}}</math>
::<math> {v_{f}}={9m/s}</math>

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===
http://hyperphysics.phy-astr.gsu.edu/hbase/colsta.html
http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Momentum-Real-life-applications.html

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]