Physics Book - User contributions [en]

Magnetic Field of a Loop

2018-04-18T22:22:21Z

Nharris41:

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

[[File: Mag Field in Loop.PNG|thumb|250px|Image depicting the magnetic field coming from a loop with a current.]]

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule. Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

The other version of the right-hand rule, which will help you when determining the direction of the magnetic field is as follows: The thumb of the right hand points in the direction of conventional current flow, I, and the fingers curl around in the direction of the magnetic field. The reverse is true as well, with the fingers curling in the direction of conventional current, and the thumb pointing in the direction of the magnetic field, as the right hand rule is essentially a depiction of the cross product between the two values.

[[File: RHR_magneticfield.png]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.png]]
a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 

a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}}(10) </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3}(10) </math>

= 2.98e-06 T

*note, that since R <<< z, the approximate formula would be viable in this case for computing magnetic field, in an alternate problem.

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T22:20:49Z

Nharris41:

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

[[File: Mag Field in Loop.PNG|thumb|left|250px|Image depicting the magnetic field coming from a loop with a current.]]

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule. Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

The other version of the right-hand rule, which will help you when determining the direction of the magnetic field is as follows: The thumb of the right hand points in the direction of conventional current flow, I, and the fingers curl around in the direction of the magnetic field. The reverse is true as well, with the fingers curling in the direction of conventional current, and the thumb pointing in the direction of the magnetic field, as the right hand rule is essentially a depiction of the cross product between the two values.

[[File: RHR_magneticfield.png]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.png]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 

a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}}(10) </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3}(10) </math>

= 2.98e-06 T

*note, that since R <<< z, the approximate formula would be viable in this case for computing magnetic field, in an alternate problem.

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T22:13:50Z

Nharris41: /* Direction of Magnetic Field */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule. Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

The other version of the right-hand rule, which will help you when determining the direction of the magnetic field is as follows: The thumb of the right hand points in the direction of conventional current flow, I, and the fingers curl around in teh direction of the magnetic field.

[[File: RHR_magneticfield.png]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.png]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 

a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}}(10) </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3}(10) </math>

= 2.98e-06 T

*note, that since R <<< z, the approximate formula would be viable in this case for computing magnetic field, in an alternate problem.

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

File:RHR magneticfield.png

2018-04-18T22:13:22Z

Nharris41:

Magnetic Field of a Loop

2018-04-18T22:13:05Z

Nharris41: /* Direction of Magnetic Field */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule. Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

The other version of the right-hand rule, which will help you when determining the direction of the magnetic field is as follows: The thumb of the right hand points in the direction of conventional current flow, I, and the fingers curl around in teh direction of the magnetic field.

[[File: RHR magneticfield.png]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.png]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 

a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}}(10) </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3}(10) </math>

= 2.98e-06 T

*note, that since R <<< z, the approximate formula would be viable in this case for computing magnetic field, in an alternate problem.

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T22:00:06Z

Nharris41: /* Examples */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule as follows:

Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field.
For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.png]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 

a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}}(10) </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3}N </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3}(10) </math>

= 2.98e-06 T

*note, that since R <<< z, the approximate formula would be viable in this case for computing magnetic field, in an alternate problem.

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T21:46:55Z

Nharris41: /* Examples */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule as follows:

Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field.
For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.png]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 

a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}} </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3} </math>

= 2.98e-06 T

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T21:44:47Z

Nharris41:

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule as follows:

Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field.
For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.PNG]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 
a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{((0.3 m)^2 + (0.02 m)^2)^{3/2}} </math>

= 2.90e-06 T

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3} </math>

= 2.98e-06 T

c) If the current in the right loop was counterclockwise, the magnetic field would point to the left, towards the left coil (determined via the right hand rule). this means that the left coil and the right coil would have opposing magnetic fields, and since they are identical coils, that would make the magnitude of the magnetic force at x, 0.

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T21:37:09Z

Nharris41: /* Example */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule as follows:

Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field.
For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Examples===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

[[File: Magneticfieldofcoils.PNG]]

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

 ANSWER 2 
a) Using the right hand rule, when viewed from the right side, we know that the since current runs counter-clockwise as viewed from the right side, the magnetic field from both coils is pointing to the right. So, to calculate the magnetic field at the midpoint, we need only use the magnetic field from the left coil. To clarify, the direction of the magnetic field is to the right, or in the (+x) direction.

To calculate the magnitude of the magnetic field of a loop, we will use the full formula:

= 2.90e-06

b) Using the approximate formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(2 A) \pi (0.02 m)^2}{(0.3 m)^3} </math>

= 2.98e-06 T

c)

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

File:Magneticfieldofcoils.png

2018-04-18T21:15:44Z

Nharris41: Image depicting example problem for magnetic field of a loop

Image depicting example problem for magnetic field of a loop

Magnetic Field of a Loop

2018-04-18T21:12:19Z

Nharris41: /* Direction of Magnetic Field */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

This image depicts the magnetic field coming from a loop with a current.

The direction of the magnetic field at a given point near the loop can be determined using the right hand rule as follows:

Point your fingers of your right hand in the direction of the conventional current, and curl them over the vector r will allow your thumb to point in the direction of the magnetic field.
For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Example===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-04-18T21:04:41Z

Nharris41: /* Example */

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

The right hand rule can be used to find the direction of the magnetic field at a given point. Putting your right fingers in the direction of the conventional current, and curling them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Example===
 QUESTION 1 
Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 1 

Using the equation for magnetic field of a loop,

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

you can simply plug in the numbers to get:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

 QUESTION 2 
Two thin coils of radius R = 2 cm are d = 30 cm apart and concentric with a common axis. Both coils contain 10 turns of wire with a conventional current of I = 2 amperes that runs counter-clockwise as viewed from the right side, as shown on the figure below.

a) What is the magnitude and direction of the magnetic field on the axis, halfway between the two loops, without making the approximation z >> r?

b) In this situation, the observation location is not very far from either coil. Calculate the magnitude of the magnetic field at the same location, using the 1/z3 approximation.

c) What is the magnitude and direction of the magnetic field midway between the two coils if the current in the right loop is reversed to run clockwise?

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

Magnetic Field of a Loop

2018-03-06T07:12:54Z

Nharris41:

Claimed by Jeffrey Mullavey; CLAIMED BY JORDAN MARSHALL TO EDIT FALL 2016
Claimed by Qianyi Qu Spring 2017
'''Claimed by Nicole Harris Spring 2018'''
== Creation of a Magnetic Field around a Current-Carrying Loop==

Like other magnetic field patterns, a magnetic field can be created through motion of charge through a loop. The system is not considered to be in equilibrium, therefore there is a movement of a mobile sea of electrons, which causes an electric current in the wire. Ideal loops are considered to be circular, so for the sake of calculations, a perfectly circular loop will be used. Thus, the conventional current is directed clockwise or counterclockwise through the loop, and depending on the direction of the flow of current, the magnetic field on the axis through the center of the loop will either go in the positive or negative direction of the axis, as shown below. Formulas have been derived to assist in the calculation of these magnetic fields.

==Calculation of Magnetic Field==

The magnetic field created by a loop is easiest to calculate on axis. This means drawing a line though the center of the loop perpendicular to the plane of the loop. This axis is commonly referred to as the "z-axis." The magnetic field is calculated by integrating across the bounds of the loop (0 to 2 pi), but can also be approximated with great accuracy using a derived formula. Calculation of magnetic field off of this axis is much more difficult, and usually requires the assistance of computer software. For this reason, only the calculation on axis will be addressed.

===Magnitude of Magnetic Field===

When calculating the magnetic field at a point on the z-axis, one can use the following formula:

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>

This allows for the calculation of the magnitude in units of Teslas.

If the distance from the center of the loop is much greater than the radius of the loop, an approximation can be made. The formula can be simplified to

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} \text{ ,where z is much greater than R} </math>

===Direction of Magnetic Field===

[[File: Mag Field in Loop.PNG]]

The right hand rule can be used to find the direction of the magnetic field at a given point. Putting your right fingers in the direction of the conventional current, and curling them over the vector r will allow your thumb to point in the direction of the magnetic field. For example, a conventional current, I, running counterclockwise would produce a field pointing out of the page on the z-axis. This rule holds regardless of where the observational location is on the center axis.

[[File: RHR7.PNG]]

The magnetic field pattern for locations outside the ring points in the opposite direction, in accordance with the right hand rule. The right fingers still point in the direction of the current, however the observational vector perpendicular to the perimeter of the loop is now in the opposite direction. Thus, the magnetic field is in the opposite direction.

===Multiple Loops===

To provide a higher magnitude in the magnetic field, many practical applications require the addition of multiple loops to "magnify" the effect of the magnetic field. In many scenarios, electric current will run through a formation of a number of loops, N, to satisfy this increasing need for a larger magnetic field. In this case, the magnitude of the induced magnetic field can be found by calculating the field produced by one loop and multiplying it by the number of loops.

This equation for the multiple loop configuration of the magnetic field of a loop will follow the equation below.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{(z^2 + R^2)^{3/2}} N </math>

===Example===

Point C is located 2 meters away from the center of a loop of current. The loop has 5 Amps of current flowing around it and has a radius of .1 meters. What is the magnetic field at point C?

 ANSWER 

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2I \pi R^2}{z^3} </math>

Using this equation, you can plug in the numbers.

<math>\vec B=\frac{\mu_0}{4 \pi} \frac{2(5) \pi (.1)^2}{2^3} </math>

B = 3.93e-9 T

==Connectedness==
Magnetic fields from electric loops are observed often in science and industrial applications. For example, a solenoid is often modeled as a bunch of loops contributing to a collective magnetic field. It is important to be able to model the field, since many scientific applications of magnetism involve circular loops. The calculations can be compared to experiments done in the lab.

This validation of theory can be extended to industrial and technological innovations seen throughout different career paths. Take, for example, the thriving CubeSat business. CubeSats are essentially miniaturized satellites (sometimes weighing between 8 and 4 kg) that perform very specific scientific missions. The payloads carried on these satellites often require a specific attitude with respect to the Earth/Sun in order to complete communications with a ground station below or to continue to receive solar power from the sun. To do this, many CubeSats often employ the use of Magnetorquer rods. These rods are essentially a a solenoid (i.e. current-carrying loop with several thousand loops, N) that surrounds a metal rod (sometimes Stainless Steel). When current flows through the coils, a magnetic field is induced along the axis of the rod. The magnetic field continues to interact with the Earth's magnetic field, producing a torque on the spacecraft. When multiple magnetorquer rods are used on different axes of the spacecraft, the CubeSat can have complete control it's attitude to continue the mission as necessary.

In all concentrations of engineering, electric and magnetic properties are important. For example, these concepts can be applied to the synthesis and manufacturing of conductors and carbon nanotubes. In many electrical experiments, it is important to understand how objects will be impacted by the creation of an induced magnetic field.

==See Also==

There are quite a few youtube videos out there that may explain the process for calculating the magnetic field on the axis through a loop carrying a current. One of those such videos can be found [https://www.youtube.com/watch?v=lN296gUXkl4 here]

==References==

Matter and Interactions Vol. II

[[Category:Fields]]

File:EarthTranslationa3.jpg

2017-11-30T01:07:41Z

Nharris41:

File:TransMiddle3.jpg

2017-11-30T00:23:57Z

Nharris41:

File:HardTranslational2.jpg

2017-11-30T00:23:36Z

Nharris41:

File:TransMiddle2.jpg

2017-11-29T23:50:28Z

Nharris41:

Entropy

2017-11-29T21:28:07Z

Nharris41:

==The Main Idea==

Entropy is an important idea as it is crucial to both the fields of physics and chemistry, but often times it is hard to understand. The traditional definition of entropy is "the degree of disorder or randomness in the system" (Merriam). This definition can however can get lost on some people. A good way to visualize how entropy works is to think of it as a probability distribution with energy. In a sample space which includes two models and 8 quanta, you can configure each quanta to any system you like. All 8 quanta could go to one system, or they can be evenly distributed. If there each systems have equal probabilities of quanta levels, then a whole distribution can be formed around it. In this model, the probability that the energy will reach equilibrium is the highest, while scenarios where all the quanta is located in exclusively one of the two models have the lowest probability. In this way the new definition of entropy becomes "the direct measure of each energy configuration's probability."

===A Mathematical Model===

Here is a formula to calculate how many ways there are to arrange q quanta among n one-dimensional oscillators:

[[File:Example.jpg]]

From this you can directly calculate Entropy (S):

[[File:Example.jpg]]

Where (The Boltzmann constant) Kb = 1.38 e -23

===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==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==

In my research I read that entropy is known as time's arrow, which in my opinion is one of the most powerful denotations of a physics term. Entropy is a fundamental law that makes the universe tick and it is such a powerful force that it will (possibly) cause the eventual end of the entire universe. Since entropy is always increases, over the expanse of an obscene amount of time the universe due to entropy will eventually suffer a "heat death" and cease to exist entirely. This is merely a scientific hypothesis, and though it may be gloom, an Asimov supercomputer Multivac may finally solve the Last Question and reboot the entire universe again.

The study of entropy is pertinent to my major as an Industrial Engineer as the whole idea of entropy is statistical thermodynamics. This is very similar to Industrial Engineering as it is essentially a statistical business major. Though the odds are unlikely that entropy will be directly used in the day of the life of an Industrial Engineer, the same distributions and concepts of probability are universal and carry over regardless of whether the example is of thermodynamic or business.

My understanding of quantum computers is no more than a couple of wikipedia articles and youtube videos, but I assume anything along the fields of quantum mechanics, which definitely relates to entropy, is important in making the chips to withstand intense heat transfers, etc.

==History==

The first person to give entropy a name was Rudolf Clausius. He questioned in his work the amount of usable heat lost during a reaction, and contrasted the previous view held by Sir Isaac Newton that heat was a physical particle. Clausius picked the name entropy as in Greek en + tropē means "transformation content."

The concept of Entropy was then expanded on by mainly Ludwig Boltzmann who essentially modeled entropy as a system of probability. Boltzmann gave a larger scale visualization method of an ideal gas in a container; he then stated that the logarithm of each of the micro-states each gas particle could inhabit times the constant he found was the definition of Entropy.

In this way Entropy came from an idea expounding terms of thermodynamics to a statistical thermodynamics which has many formulas and ways of calculation.

== See also ==

Here are a list of great resources about entropy that make it easier to understand, and also help expound more on the details of the topic.

===Further reading===

===External links===

Great TED-ED on the subject:
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips

==References==

*http://gallica.bnf.fr/ark:/12148/bpt6k152107/f369.table
*http://www.panspermia.org/seconlaw.htm
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips
*https://www.merriam-webster.com/dictionary/entropy

[[Category:Which Category did you place this in?]]

Real Systems

2017-11-29T20:55:20Z

Nharris41:

==The Main Idea==
[[File:RealPointParticleDifference.PNG|thumb|left]]
In [[Point Particle Systems]], the only change in energy is from translational kinetic energy because every force is assumed to act on the center of mass. Up until Week 10, we have been measuring change in energy of systems using the Point Particle Method. From what we learned in Week 10 though, we know that translational kinetic energy is not the only type of energy there can be a change in (see: [[Thermal Energy]] and [[Translational, Rotational and Vibrational Energy]]). In a real system, you must consider the point of application of each force when calculating the change in energy. Also in real systems, forces may also occur over a different displacement than the displacement of the center of mass. These two key differences lead to an interesting mathematical model that differs from that used for the Point Particle Method. Another distinguishing difference is that the point particle method looks at a system's center of mass and its movement. Real systems consider the change in distance with regard to the point of contact.

===A Mathematical Model===

The mathematical equation used for Real Systems can vary depending on what his happening within and on the system. For the sake of flow with the WikiPhysicsBook, we will be analyzing real systems with the energy principle.

[[File:EnergyPrinEqn.png]]>

(We are ignoring Q for the sake of simplicity. It will not be taken into account in the subsequent examples despite the possible transfer of energy from temperature differences).
'''E''' is the total energy of the system and '''W''' is the net work done from the surroundings on system. The major difference of a point particle system versus a real system is in the calculation of Work. In a point particle system, it is calculated by the net force dot product with the change in the position of the center of mass. However, Work in a real system is calculated by:

[[File:Workeq.PNG]]

This means that the summation of the all the external forces dot product with the distance each force was applied amounts to the total change in energy of the real system. The change in the mathematical equation for Work between a point particle system and a real system is important because now different forms of energy may be taken into account. In a real system, the change in energy of a system can be given by:

[[File:Realenergyeq.PNG]]

Where,
total change in internal energy ('''U''') is given by:

[[File:Utotaleq.PNG]]

total change in kinetic energy ('''K''') is given by:

[[File:Ktoteq.PNG]]

and change in Miscellaneous Energy is given by:

[[File:Emisceq.PNG]]

===A Computational Model===

==Examples==

In order to better display the difference of Real Systems from [[Point Particle Systems]], the examples done here will be the same examples done from [[Point Particle Systems]].

===Jumper Model (Simple)===
'''
'''Problem:''' You jump up so that your center of mass has moved a distance '''h'''. How much chemical energy did you expend?

From the Point Particle System analysis, we know that [[File:Jumpktrans.PNG]] and [[File:Fnetjump.PNG]].

System: Person Surroundings: Earth+Floor

Initial State: Crouched down

Final State: Extended and moving with speed v

[[File:Jumpsteps.PNG]]

Assuming negligent change in thermal energy and relative kinetic energy, the change in thermal energy is approximately equal to the normal force multiplied by height.

===Yo-Yo (Middling)===

[[File:Simple.png|650px]][http://www.example.com link title](Chabay)

'''Step 1:''' Solve for translational kinetic energy using the Point Particle System

(The equation for translational kinetic energy here is different than that in [[Point Particle Systems]], so the derivation has been provided.)

[[File:Middling.png|650px]]

[[File:Simple Part Two.png|650px]]

'''Step Two:''' Solve for rotational kinetic energy using a Real System

[[File:Difficult.png]]

Here is where the true difference between Real and Point Particle Systems can be seen. In the Point Particle system, there is no value to account for the change of rotational kinetic energy from the work done the hand. By changing the Work equation to [[File:Workeq.PNG]] rather than [[File:Wppeq.PNG]], the rotational kinetic energy can now be found.

===Spring In a Box (Difficult)===

Suppose a thin box contains a ball of clay with the mass '''M''' connected to a relaxed spring with a stiffness '''ks'''. The masses of the box and the spring are negligible. It is initally at rest, and then a constant force of '''F'''. The box moves a distance '''b''' and the spring stretches a distance '''s''' so that the clay sticks to the box. What is the change in thermal energy of the clay after colliding with the wall of the box?

[[File:Springbox.png]]

From the analysis of the [[Point Particle Systems]] of the Spring in a Box, we know that [[File:Ktransbox.PNG]]. Because the system is a spring, we also know that [[File:Uspring.PNG]].

[[File:Bssol.PNG]]

Assuming there is no relative kinetic energy (none based on diagram) and no change in chemical energy (there is no change in substance), the change in thermal energy of the clay can be found. Finding the change in thermal energy is important because you can determine whether there was enough energy to change the temperature of the clay or whether there is enough energy given off by the clay to change the temperature of a surrounding substance by a degree. Problems like this show the importance of analyzing real systems versus point particle systems.

==Connectedness==
'''How is this topic connected to something that you are interested in?'''

This topic interests me because from one single system you can mathematically determine the other forms of energy that can occur in various physical interaction. From the other forms of energy, you can determine whether there is enough energy to maybe change the temperature of another substance via thermal energy or even change the substance that is in the system given a big enough change in chemical energy.

'''How is it connected to your major?'''

As a chemical engineering major, the application of Real Systems is largely used for the majority of mathematics in my major dealing with energy balances. From only analyzing a system from a point particle method, one would only be able to find the change in the translational kinetic energy. In my major, it is very important to consider the entire system in order to find important values such as the change in thermal and kinetic energy because these values are often associated with the amount of work and heat produced in many chemical engineering processes.

'''Is there an interesting industrial application?'''

There is an absolute overload of interesting industrial applications for the analysis of real systems. In fact, the analysis of real systems in terms of energy balances is the entirety of what I've done in my chemical engineering classes thus far (I am currently a second year). There are many interesting (depending on your taste) uses of the real system analysis on a multitude of different turbines and chemical reactors.

==See also==

===Further reading===
http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:pp_vs_real

===External links===
[http://www.physicsbook.gatech.edu/Point_Particle_Systems]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. "9." Matter & Interactions. N.p.: n.p., n.d. N. pag. Print.

Wiki Commons Picture

--[[User:Nfortingo3|Nfortingo3]]([[User talk:Nfortingo3|talk]]) 19:26, 28 November 2015 (EST)

[[Category:Energy]]

Entropy

2017-11-29T20:54:18Z

Nharris41:

'''Claimed by Nicole Harris Fall 2017'''

==The Main Idea==

Entropy is an important idea as it is crucial to both the fields of physics and chemistry, but often times it is hard to understand. The traditional definition of entropy is "the degree of disorder or randomness in the system" (Merriam). This definition can however can get lost on some people. A good way to visualize how entropy works is to think of it as a probability distribution with energy. In a sample space which includes two models and 8 quanta, you can configure each quanta to any system you like. All 8 quanta could go to one system, or they can be evenly distributed. If there each systems have equal probabilities of quanta levels, then a whole distribution can be formed around it. In this model, the probability that the energy will reach equilibrium is the highest, while scenarios where all the quanta is located in exclusively one of the two models have the lowest probability. In this way the new definition of entropy becomes "the direct measure of each energy configuration's probability."

===A Mathematical Model===

Here is a formula to calculate how many ways there are to arrange q quanta among n one-dimensional oscillators:

[[File:Example.jpg]]

From this you can directly calculate Entropy (S):

[[File:Example.jpg]]

Where (The Boltzmann constant) Kb = 1.38 e -23

===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==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==

In my research I read that entropy is known as time's arrow, which in my opinion is one of the most powerful denotations of a physics term. Entropy is a fundamental law that makes the universe tick and it is such a powerful force that it will (possibly) cause the eventual end of the entire universe. Since entropy is always increases, over the expanse of an obscene amount of time the universe due to entropy will eventually suffer a "heat death" and cease to exist entirely. This is merely a scientific hypothesis, and though it may be gloom, an Asimov supercomputer Multivac may finally solve the Last Question and reboot the entire universe again.

The study of entropy is pertinent to my major as an Industrial Engineer as the whole idea of entropy is statistical thermodynamics. This is very similar to Industrial Engineering as it is essentially a statistical business major. Though the odds are unlikely that entropy will be directly used in the day of the life of an Industrial Engineer, the same distributions and concepts of probability are universal and carry over regardless of whether the example is of thermodynamic or business.

My understanding of quantum computers is no more than a couple of wikipedia articles and youtube videos, but I assume anything along the fields of quantum mechanics, which definitely relates to entropy, is important in making the chips to withstand intense heat transfers, etc.

==History==

The first person to give entropy a name was Rudolf Clausius. He questioned in his work the amount of usable heat lost during a reaction, and contrasted the previous view held by Sir Isaac Newton that heat was a physical particle. Clausius picked the name entropy as in Greek en + tropē means "transformation content."

The concept of Entropy was then expanded on by mainly Ludwig Boltzmann who essentially modeled entropy as a system of probability. Boltzmann gave a larger scale visualization method of an ideal gas in a container; he then stated that the logarithm of each of the micro-states each gas particle could inhabit times the constant he found was the definition of Entropy.

In this way Entropy came from an idea expounding terms of thermodynamics to a statistical thermodynamics which has many formulas and ways of calculation.

== See also ==

Here are a list of great resources about entropy that make it easier to understand, and also help expound more on the details of the topic.

===Further reading===

===External links===

Great TED-ED on the subject:
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips

==References==

*http://gallica.bnf.fr/ark:/12148/bpt6k152107/f369.table
*http://www.panspermia.org/seconlaw.htm
*https://ed.ted.com/lessons/what-is-entropy-jeff-phillips
*https://www.merriam-webster.com/dictionary/entropy

[[Category:Which Category did you place this in?]]

Gyroscopes

2017-11-28T17:54:51Z

Nharris41:

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained further on in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

'''Lrot,r = Iω'''

Where the rotational angular momentum points horizontal to the rotor.

The Lrot,r will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

'''Ω = ''V''cm/''r'''''

Where ''r'' is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''Lsupport = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

'''LrotΩ'''

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

'''Ω = τCM/Lrot = ''r''Mg/''I''ω '''

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes offer two functions in aircraft. The first is rigidity in space, which means that the gyroscope will resist any attempt to change the direction of the axis. This is useful when the plane makes turns, throwing the plane off its natural horizontal. The gyroscope acts as a "fake horizon", which orientates the plane back to its natural position. The technical term for this gyroscope contraption is an attitude indicator.

The second function of gyroscopes in planes is precession. In this context, this means that any perpendicular force applied to a gyroscope's axis of rotation will manifest itself 90° further along the axis of rotation. This property is useful because the amount of banking before a turn can be determined. If the gyroscope was oriented with the longitudinal axis of the plane, then only the rate of turn could be determined instead of the amount of bank on the aircraft.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]

Real Systems

2017-11-28T17:54:13Z

Nharris41:

'''Claimed by Nicole Harris 2017'''

==The Main Idea==
[[File:RealPointParticleDifference.PNG|thumb|left]]
In [[Point Particle Systems]], the only change in energy is from translational kinetic energy because every force is assumed to act on the center of mass. Up until Week 10, we have been measuring change in energy of systems using the Point Particle Method. From what we learned in Week 10 though, we know that translational kinetic energy is not the only type of energy there can be a change in (see: [[Thermal Energy]] and [[Translational, Rotational and Vibrational Energy]]). In a real system, you must consider the point of application of each force when calculating the change in energy. Also in real systems, forces may also occur over a different displacement than the displacement of the center of mass. These two key differences lead to an interesting mathematical model that differs from that used for the Point Particle Method. Another distinguishing difference is that the point particle method looks at a system's center of mass and its movement. Real systems consider the change in distance with regard to the point of contact.

===A Mathematical Model===

The mathematical equation used for Real Systems can vary depending on what his happening within and on the system. For the sake of flow with the WikiPhysicsBook, we will be analyzing real systems with the energy principle.

[[File:EnergyPrinEqn.png]]>

(We are ignoring Q for the sake of simplicity. It will not be taken into account in the subsequent examples despite the possible transfer of energy from temperature differences).
'''E''' is the total energy of the system and '''W''' is the net work done from the surroundings on system. The major difference of a point particle system versus a real system is in the calculation of Work. In a point particle system, it is calculated by the net force dot product with the change in the position of the center of mass. However, Work in a real system is calculated by:

[[File:Workeq.PNG]]

This means that the summation of the all the external forces dot product with the distance each force was applied amounts to the total change in energy of the real system. The change in the mathematical equation for Work between a point particle system and a real system is important because now different forms of energy may be taken into account. In a real system, the change in energy of a system can be given by:

[[File:Realenergyeq.PNG]]

Where,
total change in internal energy ('''U''') is given by:

[[File:Utotaleq.PNG]]

total change in kinetic energy ('''K''') is given by:

[[File:Ktoteq.PNG]]

and change in Miscellaneous Energy is given by:

[[File:Emisceq.PNG]]

===A Computational Model===

==Examples==

In order to better display the difference of Real Systems from [[Point Particle Systems]], the examples done here will be the same examples done from [[Point Particle Systems]].

===Jumper Model (Simple)===
'''
'''Problem:''' You jump up so that your center of mass has moved a distance '''h'''. How much chemical energy did you expend?

From the Point Particle System analysis, we know that [[File:Jumpktrans.PNG]] and [[File:Fnetjump.PNG]].

System: Person Surroundings: Earth+Floor

Initial State: Crouched down

Final State: Extended and moving with speed v

[[File:Jumpsteps.PNG]]

Assuming negligent change in thermal energy and relative kinetic energy, the change in thermal energy is approximately equal to the normal force multiplied by height.

===Yo-Yo (Middling)===

[[File:Simple.png|650px]][http://www.example.com link title](Chabay)

'''Step 1:''' Solve for translational kinetic energy using the Point Particle System

(The equation for translational kinetic energy here is different than that in [[Point Particle Systems]], so the derivation has been provided.)

[[File:Middling.png|650px]]

[[File:Simple Part Two.png|650px]]

'''Step Two:''' Solve for rotational kinetic energy using a Real System

[[File:Difficult.png]]

Here is where the true difference between Real and Point Particle Systems can be seen. In the Point Particle system, there is no value to account for the change of rotational kinetic energy from the work done the hand. By changing the Work equation to [[File:Workeq.PNG]] rather than [[File:Wppeq.PNG]], the rotational kinetic energy can now be found.

===Spring In a Box (Difficult)===

Suppose a thin box contains a ball of clay with the mass '''M''' connected to a relaxed spring with a stiffness '''ks'''. The masses of the box and the spring are negligible. It is initally at rest, and then a constant force of '''F'''. The box moves a distance '''b''' and the spring stretches a distance '''s''' so that the clay sticks to the box. What is the change in thermal energy of the clay after colliding with the wall of the box?

[[File:Springbox.png]]

From the analysis of the [[Point Particle Systems]] of the Spring in a Box, we know that [[File:Ktransbox.PNG]]. Because the system is a spring, we also know that [[File:Uspring.PNG]].

[[File:Bssol.PNG]]

Assuming there is no relative kinetic energy (none based on diagram) and no change in chemical energy (there is no change in substance), the change in thermal energy of the clay can be found. Finding the change in thermal energy is important because you can determine whether there was enough energy to change the temperature of the clay or whether there is enough energy given off by the clay to change the temperature of a surrounding substance by a degree. Problems like this show the importance of analyzing real systems versus point particle systems.

==Connectedness==
'''How is this topic connected to something that you are interested in?'''

This topic interests me because from one single system you can mathematically determine the other forms of energy that can occur in various physical interaction. From the other forms of energy, you can determine whether there is enough energy to maybe change the temperature of another substance via thermal energy or even change the substance that is in the system given a big enough change in chemical energy.

'''How is it connected to your major?'''

As a chemical engineering major, the application of Real Systems is largely used for the majority of mathematics in my major dealing with energy balances. From only analyzing a system from a point particle method, one would only be able to find the change in the translational kinetic energy. In my major, it is very important to consider the entire system in order to find important values such as the change in thermal and kinetic energy because these values are often associated with the amount of work and heat produced in many chemical engineering processes.

'''Is there an interesting industrial application?'''

There is an absolute overload of interesting industrial applications for the analysis of real systems. In fact, the analysis of real systems in terms of energy balances is the entirety of what I've done in my chemical engineering classes thus far (I am currently a second year). There are many interesting (depending on your taste) uses of the real system analysis on a multitude of different turbines and chemical reactors.

==See also==

===Further reading===
http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:pp_vs_real

===External links===
[http://www.physicsbook.gatech.edu/Point_Particle_Systems]

==References==

Chabay, Ruth W., and Bruce A. Sherwood. "9." Matter & Interactions. N.p.: n.p., n.d. N. pag. Print.

Wiki Commons Picture

--[[User:Nfortingo3|Nfortingo3]]([[User talk:Nfortingo3|talk]]) 19:26, 28 November 2015 (EST)

[[Category:Energy]]

Gyroscopes

2017-11-27T03:45:45Z

Nharris41:

'''claimed by Nicole Harris Fall 2017'''

An explanation by Ansley Marks

A [https://en.wikipedia.org/wiki/Gyroscope gyroscope] is a device containing a wheel or disk that is free to rotate about its own axis independent of a change in direction of the axis itself. Since the spinning wheel persists in maintaining its plane of rotation, a [https://www.youtube.com/watch?v=ty9QSiVC2g0 gyroscopic effect] can be observed.

[[File: gyro.gif]]

==The Main Idea==

Although insignificant looking and seemingly uninteresting when still, gyroscopes become a fascinating device when in motion and can be explained using the angular momentum principle. Gyroscopes come in all different forms with varying parts. The main component of a gyroscope is a spinning wheel or a disk mounted on an axle. Typically gyroscopes contain a suspended rotor inside three rings called gimbals. In order to ensure that little torque is applied to the inside rotor, the gimbals are mounted on high quality bearing surfaces, allowing free movement of the spinning wheel in the middle. These types of gyroscopes with multiple gimbals are useful for stabilization because the wheels can change direction without affecting the inner rotor. If the spinning axle of a gyroscope is placed on a support, then a complex motion can be observed. The motion of a gyroscope will be modeled and explained further on in this page.

===A Mathematical Model===

When the spinning axis of a gyroscope is placed on a support, a gyroscopic effect is observed. The gyroscope bobs up and down--nutation--and rotates about the support--precession. For the sake of simplifying the mathematical equations for a gyroscope's motion, [http://dictionary.reference.com/browse/nutation nutation] (the upwards and downwards movement of the rotor) will be ignored. We will only look at the [http://dictionary.reference.com/browse/precession precession] motion of the gyroscope.

[[File:gyropic1.png|200px|thumb|left|A gyroscope processing in the x,z plane with the y-axis positioned upwards along the vertical support.]]

To start off with, the gyroscope's rotor rotates about its own axis with an angular velocity of ω and has a moment of inertia ''I''. Thus, the rotational angular momentum of the rotor can be modeled as:

'''Lrot,r = Iω'''

Where the rotational angular momentum points horizontal to the rotor.

The Lrot,r will always change direction as the rotor rotates about the support. The rotor processes about the support with an angular velocity Ω, which is constant in magnitude and direction.

If Ω is known, then the velocity of the center of mass of the rotor device can be derived using the following relationship:

'''Ω = ''V''cm/''r'''''

Where ''r'' is equal to the distance from the support to the center of mass of the rotor device. The linear momentum of the gyroscope is then Ω''P''. [[File:figure11.611.png|200px|thumb|left|A view of the gyroscope from the side with all the forces labeled.]]

Since the rotor is processing about the support, there must be a perpendicular force ''f'' exerted by the support such that Ω''P'' = ''f'', where ''P'' is equal to ''M(Ωr)''. Thus, ''f'' = ''Mr''<math>Ω^2</math>.

There is also a translational angular momentum of the rotor processing about the support. This can be modeled by finding the magnitude of the position vector crossed with the momentum.

'''Lsupport = |''R'' x ''P'' | '''

Since the direction of the rotational angular momentum of the rotor around the support is constantly changing direction, the rate of change of the rotational angular momentum can be written as:

'''LrotΩ'''

Thus the only remaining element that is needed to complete the Angular Momentum Principle is the torque. The torque is equal to the distance from the support to the center of mass of the rotor, ''r'', multiplied by the force exerted, which is the mass times gravity. Therefore, since the change in rotational angular momentum is ''LrotΩ'', that must be equal to ''τCM''. By setting the two equations equal to each other, the angular momentum can be isolated to one side. This yields the following result:

'''Ω = τCM/Lrot = ''r''Mg/''I''ω '''

==Real World Examples==

===Magnetic Resonance Imaging===

A good analogy for the way that a Magnetic Resonance Imaging (MRI) works is a gyroscope. To start off with a little background, the way that an MRI works is that all the hydrogen atoms in your body are aligned by using strong magnetic fields. Once these hydrogen atoms are aligned, similar to how a compass's needle is aligned, radio waves can be sent into the body and signals are created from the way the photons emit the radio waves.

Identical to gyroscopes, the hydrogen nucleus rotates about its own axis at a particular frequency. The strength and direction of the magnetic field can effect the direction and angular speed of these rotating protons in the nuclei. By controlling the direction and rotation speed, the location of the hydrogen nucleus can be deduced and thus helping the process of creating images.

===Aviation===

Gyroscopes offer two functions in aircraft. The first is rigidity in space, which means that the gyroscope will resist any attempt to change the direction of the axis. This is useful when the plane makes turns, throwing the plane off its natural horizontal. The gyroscope acts as a "fake horizon", which orientates the plane back to its natural position. The technical term for this gyroscope contraption is an attitude indicator.

The second function of gyroscopes in planes is precession. In this context, this means that any perpendicular force applied to a gyroscope's axis of rotation will manifest itself 90° further along the axis of rotation. This property is useful because the amount of banking before a turn can be determined. If the gyroscope was oriented with the longitudinal axis of the plane, then only the rate of turn could be determined instead of the amount of bank on the aircraft.

==Connectedness==
This topic is interesting because gyroscopes have held the fascination of pretty much anyone that has ever seen one in motion including myself. Although the explanation that I gave was a simplified version of a gyroscope which only processes and doesn't nutate, there are many other complex mathematical models of the complicated motion of gyroscopes. Many papers and even books have been written on the subject of gyroscopes, and they have baffled nobel prize winners such as Niels Bohr and famous physicists alike. Gyroscopes are connected to my major because they are huge in industrial manufacturing of numerous materials. We use some sort of gyroscope in our everyday lives from cars to airplanes and other mechanical equipments.

==History==

Gyroscopes have been around for nearly 200 years. The first person to discover the gyroscope was Johann Bohnenberger in 1817 at the University of Tubingen. However, Bohnenberger was not credited with the discovery of the gyroscope. The French scientist Jean Bernard Leon Foucault (1826-1864) coined the term "gyroscope" and ended up with being credited for the discovery of a gyroscope. Thanks to his experiments with the gyroscope, they started to become mainstream and studied by many other physicists. In the early 20th century, gyroscopes were first used in boats and eventually in aircraft. Gyroscopes have been modified and tweaked to suit many purposes that are widely used today mainly as stabilizers.

== See also ==

===Further reading===

Compass and Gyroscope: Integrating Science and Politics for the Environment

Mathematical model for gyroscope effects: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4915651

YouTube video on gyroscope procession: https://www.youtube.com/watch?v=ty9QSiVC2g0

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

===External links===

https://www.youtube.com/watch?v=ty9QSiVC2g0

http://dictionary.reference.com/browse/precession

http://dictionary.reference.com/browse/nutation

https://en.wikipedia.org/wiki/Gyroscope

==References==

Oxford Dictionaries: http://www.oxforddictionaries.com/us/definition/american_english/gyroscope

HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/gyr.html

Wikipedia: https://en.wikipedia.org/wiki/Gyroscope

Gyroscope History: http://www.gyroscopes.org/history.asp

Science Learning: http://sciencelearn.org.nz/Contexts/See-through-Body/Sci-Media/Video/So-how-does-MRI-work

Quora: https://www.quora.com/What-the-function-of-gyroscopes-in-airplane

[[Category: Angular Momentum]]