Thin and Thick Wires: Difference between revisions

Latest revision as of 07:22, 6 December 2015

In this wiki page, you will learn about thin and thick wires and how they physically operate.

Wiki created by Ryan Keefe (rkeefe3)

The Big Picture

Let's start out by saying this: no matter how much the wire's thickness changes, if a wire has a current running through it, it will always have the same number of electrons passing through it every second. This value cannot change within the wire just because of thickness. What can change however is the magnitude of the electric field caused by the wire.

Electric Field(conceptual)

Because the electric field due to a wire can be formulated by this:

[math]\displaystyle{ E(R)=\frac{\lambda}{2 \pi \epsilon_0 R} }[/math]

Since electric field is inversely related to the radius of the wire, the thicker the wire, the less the magnitude of the wire. Funny enough though, if you take a wire (we will call this wire A) with a current and replace a section of the wire with a thinner wire (we will call this wire B) and measure the magnitude of the electric field at thick sites of the wires, we will find that wire A has a larger magnitude than wire B. Why is this? Well, analyzing wire B, one can find a large gradient of surface charge across the section of thin wire replacing the thick wire, meaning this area is acting like a funnel for electrons. This funnel means that the overall speed of the electrons going through the wire. This is true for the whole wire. As stated above, every wire has a constant number of electrons passing through it every second, also known as electron current. Because of this slow down due the thin wire, the value of [math]\displaystyle{ \lambda }[/math], which is charge per unit length, goes down making the electric field magnitude go down as well.

Electric Field(mathmatical)

Another formula for the relationship of thickness versus the magnitude of the electric field is:

[math]\displaystyle{ i=nAuE }[/math]

Where [math]\displaystyle{ i }[/math] is the electron current, [math]\displaystyle{ n }[/math] is the mobile charges per volume of wire, [math]\displaystyle{ A }[/math] is the cross sectional area of the wire, [math]\displaystyle{ u }[/math] is the electron mobility, and [math]\displaystyle{ E }[/math] is the magnitude of the electric field.

Example

Example 1

In the circuit below, all the wires are made of the same material, but a section of the wire is thin.

Which of the following are true?

The electron current is the same at every location in this circuit.
The magnitude of the electric field at location D is larger than the magnitude of the electric field at location G
The magnitude of the electric field is the same at every location in this circuit.
The magnitude of the electric field at location G is smaller in this circuit than it would be if all the wires were thick.
Fewer electrons per second pass location E than location C.
There is a large gradient of surface charge on the wire between locations C and E.
There is no surface charge at all on the wire near location G.
The electron current in this circuit is less than the electron current would be if all the wires were thick.

Let's go through the choices one at a time:

Choice 1: Is true, all wires that work have the same electron current or flow throughout.

Choice 2: Is true, because electric field strength is inversely related to the radius of the wire.

Choice 3: Is false, because there are wires of different thicknesses in the total wire.

Choice 4: Is true, because the thin wire in this one makes the electron current overall slower, than the electric field strength is also weaker with the thick wire.

Choice 5: Is false, because electron current is constant throughout the wire.

Choice 6: Is true, because of the funnel between the thick wire and thin wire that crowds up the electrons.

Choice 7: Is false, because point G is near the battery terminal.

Choice 8: Is true, because the thin wire is slowing everything up like a funnel through the wire.

Example 2

Using the picture from Example 1, let's say that the electron current is 5E18 1/s, there are 4E28 mobile electrons per cubic meter, the electron mobility is 0.00006 (m/s)(V/m), and that the radius of the thing wire is 0.2mm. What is the electric field strength around point D?

Solution: Using the equation [math]\displaystyle{ i=nAuE }[/math], one can solve this problem. Rearrange the equation and plug in the numbers.

[math]\displaystyle{ E_(D)=\frac{i}{nAU}=\frac{5*10^{18}}{4*10^{28}*\pi*(0.2*10^{-3})^{2}*0.00006}=1.66 V/m }[/math]

References

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition

http://farside.ph.utexas.edu/teaching/302l/lectures/node26.html

@@ Line 1: / Line 1: @@
-==Thermodynamics==
+In this wiki page, you will learn about thin and thick wires and how they physically operate.
-This wiki was created by Ryan Keefe (rkeefe3). This topics focuses on energy work of a system but it can only deal with a large scale response to heat in a system.  '''Thermodynamics''' is the study of the work, heat and energy of a system.  The smaller scale gas interactions can explained using the kinetic theory of gases.  There are three fundamental laws that go along with the topic of thermodynamics.  They are the zeroth law, the first law, and the second law.  These laws help us understand predict the the operation of the physical system.  In order to understand the laws, you must first understand thermal equilibrium.  [[Thermal equilibrium]] is reached when a object that is at a higher temperature is in contact with an object that is at a lower temperature and the first object transfers heat to the latter object until they approach the same temperature and maintain that temperature constantly.  It is also important to note that any thermodynamic system in thermal equilibrium possesses internal energy.
+''Wiki created by Ryan Keefe (rkeefe3)''
-===Zeroth Law===
+==The Big Picture==
-The zeroth law states that if two systems are at thermal equilibrium at the same time as a third system, then all of the systems are at equilibrium with each other.  If systems A and C are in thermal equilibrium with B, then system A and C are also in thermal equilibrium with each other.  There are underlying ideas of heat that are also important.  The most prominent one is that all heat is of the same kind.  As long as the systems are at thermal equilibrium, every unit of internal energy that passes from one system to the other is balanced by the same amount of energy passing back.  This also applies when the two systems or objects have different atomic masses or material.
+Let's start out by saying this: no matter how much the wire's thickness changes, if a wire has a current running through it, it will always have the same number of electrons passing through it every second. This value cannot change within the wire just because of thickness. What can change however is the magnitude of the electric field caused by the wire.
-====A Mathematical Model====
+===Electric Field(conceptual)===
-If A = B and A = C, then B = C
+Because the electric field due to a wire can be formulated by this:
-A = B = C
-====A Computational Model====
+<math> E(R)=\frac{\lambda}{2 \pi \epsilon_0 R}</math>
-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]
+Since electric field is inversely related to the radius of the wire, the thicker the wire, the less the magnitude of the wire. Funny enough though, if you take a wire (we will call this wire A) with a current and replace a section of the wire with a thinner wire (we will call this wire B) and measure the magnitude of the electric field at thick sites of the wires, we will find that wire A has a larger magnitude than wire B. Why is this? Well, analyzing wire B, one can find a large gradient of surface charge across the section of thin wire replacing the thick wire, meaning this area is acting like a funnel for electrons. This funnel means that the overall speed of the electrons going through the wire. This is true for the whole wire. As stated above, every wire has a constant number of electrons passing through it every second, also known as electron current. Because of this slow down due the thin wire, the value of <math>\lambda</math>, which is charge per unit length, goes down making the electric field magnitude go down as well.
-===First Law===
+===Electric Field(mathmatical)===
-The first law of thermodynamics defines the internal energy (E) as equal to the difference between heat transfer (Q) ''into'' a system and work (W) ''done by'' the system.  Heat removed from a system would be given a negative sign and heat applied to the system would be given a positive sign.  Internal energy can be converted into other types of energy because it acts like potential energy.  Heat and work, however, cannot be stored or conserved independently because they depend on the process.  This allows for many different possible states of a system to exist.  There can be a process known as the adiabatic process in which there is no heat transfer.  This occurs when a system is full insulated from the outside environment.  The implementation of this law also brings about another useful state variable, '''enthalpy'''.
+Another formula for the relationship of thickness versus the magnitude of the electric field is:
-====A Mathematical Model====
+<math>i=nAuE</math>
-E2 - E1 = Q - W
+Where <math>i</math> is the electron current, <math>n</math> is the mobile charges per volume of wire, <math>A</math> is the cross sectional area of the wire, <math>u</math> is the electron mobility, and <math>E</math> is the magnitude of the electric field.
+==Example==
+===Example 1===
+In the circuit below, all the wires are made of the same material, but a section of the wire is thin.
+[[File:MediumExampleLoopRule.jpg]]
+Which of the following are true?
+# The electron current is the same at every location in this circuit.
+# The magnitude of the electric field at location D is larger than the magnitude of the electric field at location G
+# The magnitude of the electric field is the same at every location in this circuit.
+# The magnitude of the electric field at location G is smaller in this circuit than it would be if all the wires were thick.
+# Fewer electrons per second pass location E than location C.
+# There is a large gradient of surface charge on the wire between locations C and E.
+# There is no surface charge at all on the wire near location G.
+# The electron current in this circuit is less than the electron current would be if all the wires were thick.
+Let's go through the choices one at a time:
+'''Choice 1:''' Is true, all wires that work have the same electron current or flow throughout.
+'''Choice 2:''' Is true, because electric field strength is inversely related to the radius of the wire.
+'''Choice 3:''' Is false, because there are wires of different thicknesses in the total wire.
+'''Choice 4:''' Is true, because the thin wire in this one makes the electron current overall slower, than the electric field strength is also weaker with the thick wire.
+'''Choice 5:''' Is false, because electron current is constant throughout the wire.
+'''Choice 6:''' Is true, because of the funnel between the thick wire and thin wire that crowds up the electrons.
+'''Choice 7:''' Is false, because point G is near the battery terminal.
+'''Choice 8:''' Is true, because the thin wire is slowing everything up like a funnel through the wire.
+===Example 2===
+Using the picture from Example 1, let's say that the electron current is 5E18 1/s, there are 4E28 mobile electrons per cubic meter, the electron mobility is 0.00006 (m/s)(V/m), and that the radius of the thing wire is 0.2mm. What is the electric field strength around point D?
+'''Solution:''' Using the equation <math>i=nAuE</math>, one can solve this problem. Rearrange the equation and plug in the numbers.
+<math>E_(D)=\frac{i}{nAU}=\frac{5*10^{18}}{4*10^{28}*\pi*(0.2*10^{-3})^{2}*0.00006}=1.66 V/m</math>
+==References==
+Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition
+http://farside.ph.utexas.edu/teaching/302l/lectures/node26.html