Physics Book - User contributions [en]

Bar Magnet

2016-11-28T01:16:56Z

Ycho78: Undo revision 25760 by Ycho78 (talk)

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings. Magnetic field of a bar magnet can be measured with a compass and it is strongest inside the magnet. Thus making the strongest external magnetic fields, the ones near the poles. This magnetic field is created because a bar magnet has poles -- north and south. As we know already, a magnetic north pole will attract the south pole of another magnet, and repel a north pole of another magnet. A needle of a compass itself works as a magnet in this case, thus reacting to the bar magnet.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

Here is an explanation of how to derive the equation above:

[[File:equation1.jpg]]

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

Here is an explanation of how to derive the equation above:

[[File:equation2.jpg]]

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

For a better understanding of how we can computationally visualize magnetic dipole, here is an example. Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

This code creates the following visual:
[[File:MDF1D.png]]

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
# https://www.opensourcephysics.org/items/detail.cfm?ID=12361

Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T01:14:50Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

For a better understanding of how we can computationally visualize magnetic dipole, here is an example. Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

This code creates the following visual:
[[File:MDF1D.png]]

(www.opensourcephysics.org/items/detail.cfm?ID=12361)

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
# https://www.opensourcephysics.org/items/detail.cfm?ID=12361

Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T01:12:43Z

Ycho78: /* A Mathematical Model */

Bar Magnet

2016-11-28T01:02:06Z

Ycho78: /* A Mathematical Model */

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings. Magnetic field of a bar magnet can be measured with a compass and it is strongest inside the magnet. Thus making the strongest external magnetic fields, the ones near the poles. This magnetic field is created because a bar magnet has poles -- north and south. As we know already, a magnetic north pole will attract the south pole of another magnet, and repel a north pole of another magnet. A needle of a compass itself works as a magnet in this case, thus reacting to the bar magnet.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

Here is an explanation of how to derive the equation above:

[[File:Equation.png]]

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

Here is an explanation of how to derive the equation above:

[[File:Equation2.png]]

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

For a better understanding of how we can computationally visualize magnetic dipole, here is an example. Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

This code creates the following visual:
[[File:MDF1D.png]]

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
# https://www.opensourcephysics.org/items/detail.cfm?ID=12361

Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T00:54:28Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings. Magnetic field of a bar magnet can be measured with a compass and it is strongest inside the magnet. Thus making the strongest external magnetic fields, the ones near the poles. This magnetic field is created because a bar magnet has poles -- north and south. As we know already, a magnetic north pole will attract the south pole of another magnet, and repel a north pole of another magnet. A needle of a compass itself works as a magnet in this case, thus reacting to the bar magnet.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

For a better understanding of how we can computationally visualize magnetic dipole, here is an example. Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

This code creates the following visual:
[[File:MDF1D.png]]

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
# https://www.opensourcephysics.org/items/detail.cfm?ID=12361

Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

File:MDF1D.png

2016-11-28T00:49:25Z

Ycho78: Ycho78 uploaded a new version of "File:MDF1D.png"

File:MDF1D.png

2016-11-28T00:46:46Z

Ycho78:

Bar Magnet

2016-11-28T00:46:01Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

For a better understanding of how we can computationally visualize magnetic dipole, here is an example. Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

This code creates the following visual:
[[File:MDF1D.png]]

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
# https://www.opensourcephysics.org/items/detail.cfm?ID=12361

Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T00:43:15Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

[[File:Magnetic_field_2.png]]

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T00:41:54Z

Ycho78: /* A Computational Model */

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]

[[File:Magnetic_field_2.png]]

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

File:Magnetic field 2.png

2016-11-28T00:41:12Z

Ycho78:

Bar Magnet

2016-11-28T00:40:31Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

[[File:Magnetic_field_2.png]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]
== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T00:39:16Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.png]]
== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

File:Magnet code 1.png

2016-11-28T00:38:31Z

Ycho78:

Bar Magnet

2016-11-28T00:36:50Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)

A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

Following is a program written in MATLAB to visualize magnetic dipole:
[[File:magnet_code_1.jpg]]
== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Bar Magnet

2016-11-28T00:31:13Z

Ycho78:

claimed by Samah
claimed by Yeon Jae Cho (FALL 2016)
A bar magnet creates a magnetic field, just like many other device (i.e. a current carrying wire), however, it has a different pattern of magnetic field which we will explore.

__TOC__

== '''The Main Idea''' ==
----
The main idea for this topic is to explore how a bar magnet works and the effects that it has on its surroundings.

== '''A Mathematical Model''' ==
----

In physics, it is important to keep track of your frame of reference. Treat an effect as if it is arising at the source location and ending at the observation location. The source location marks the beginning point for an effect. The result of the effect is gauged at the observation location.

Due to the fact that an observation location can either be on the axis of the magnet, or off the axis of the magnet, we have two different equations. Given a bar magnet with magnetic dipole moment μ, if the observation location is on the same axis as the magnet, assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we find that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{2\mu }{r^{3}}</math>

If the observation location is not on the axis of the bar magnet, and assuming that the distance from the observation location to the magnet is much greater than the separation distance of the two poles, we conclude that:
<math> B = \frac{\mu _{0}}{4\pi }\cdot \frac{\mu }{r^{3}}</math>

== '''A Computational Model''' ==
----
[[File:VFPt cylindrical magnet thumb.svg|thumb|left|The curly magnetic field of a bar magnet.]]

As you can see in this picture, the magnetic field of a bar magnet takes the exact same form as an electric field of a dipole. The magnetic lines flow out of the north pole of the magnet, and into the south pole of the magnet, in a curling fashion. However, the 'poles' are merely just conventions. They do not represent anything, and are terms assigned to each end, but it is true that the magnetic field will always flow out of the 'north' end. The Earth itself can also be represented by the computational model of a bar magnet; however, there are a few misconceptions about this. For starters, the magnetic North Pole is actually located near the geographic South Pole, and the magnetic South Pole is located near the geographic North Pole. Furthermore, the magnetic poles are off axis, meaning the are not directly at the top and bottom of the Earth. There is a difference of almost 1.5 degrees!
It is also interesting to note that just because this illustration depicts the bar magnet as having two distinct ends, if you were to cut the magnet down the middle, it would polarize in such a way that you would end up with two bar magnets, not a single south pole and a single north pole.

[[File:Magnet0873.png|thumb|left|The magnetic field of a bar magnet.]]

This picture depicts the magnetic field based on the dipoles of the magnet. The north end is the left side of the magnet and the south end is the right side of the magnet. The field follows the direction from the north side to the south side of the magnet.

Following is a program written in MATLAB to visualize magnetic dipole:

function h=lforce2d(n,clr)
% LFORCE2D: Generates a 2D polar plot of n-lines of force of the
% Gecentric Axial Magnetic Dipole
% Usage:
% h = LFORCE2D(N, Color)
% Input:
% N = Number of lines of force.
% Color = Line color (see PLOT)
%

% RBG [Red Blue Green]
% Written by: Dr. A. Abokhodair Date: 15/12/2005
% ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

% RBG [Red Blue Green]
% Written by: Dr. A. Abokhodair Date: 15/12/2005
% ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

d2r = pi/180;
tht = (0:5:360)'*d2r;
A = 2:n;
r = sin(tht).^2*A;
nC = size(r,2);
u = ones(1,nC);
ct = cos(tht)*u;
st = sin(tht)*u;

x = r.*st;
y = r.*ct;

figure;
R = max(x(:))/8;
axis equal;
h = plot(x,y,'r');
[nR,nC] = size(x);

for iR=2:6:floor(nR/2)
for jC=1:2:nC;
p1 = [x(iR-1,jC) -y(iR-1,jC)];
p2 = [x(iR,jC) -y(iR,jC)];
hv=plotvec(p1,p2,clr);
end
end

for iR=floor(nR/2)+2:6:nR;
for jC=1:2:nC;
p1 = [x(iR-1,jC) y(iR-1,jC)];
p2 = [x(iR,jC) y(iR,jC)];
hv=plotvec(p1,p2,clr);
end
end

hold on;
hc = fcircle(R,[0,0],clr);
hv = plotvec([0 R/2], [0 -R/2],'r');
h=plot([0 0],[R/2, -R/3],'r','LineWidth',3);

title('Magnetic Dipole Field','FontSize',15);
hold off;
grid on;

h=[h;hc];

% ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

function hh=fcircle(r,o,s)
% FCIRCLE - draws a filled cirlce
% Usage:
% h = FCIRCLE(R,O)
%

% Copyright (c) 2000-05-19 by B. Rasmus Anthin.
% Rev. 2000-10-24

phi=linspace(0,2*pi);
axis equal;
if nargin==2
h=patch(r*cos(phi)+o(1),r*sin(phi)+o(2),'');
else
h=patch(r*cos(phi)+o(1),r*sin(phi)+o(2),s);
end
set(h,'edgec',get(h,'facec'));
set(h,'user',{'fcircle',r,o});
if nargout,hh=h;end
% ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

function h=plotvec(p1,p2,linespec)
% PLOTVEC - Draws a vector.
% Usage:
% PLOTVEC(p1,p2,[LINESPEC])
%
% Draws a vector from point P1 to point P2 using linespecification
% LINESPEC (see PLOT).
%
% H = PLOTVEC(...) returns handles of the objects.
%
% See also CIRCLE, FCIRCLE, ELLIPSE, FELLIPSE.

% Copyright (c) 2001-09-28, B. Rasmus Anthin.

if nargin==2,linespec='';end
held=ishold;hold on
x=[p1(1) p2(1)];
y=[p1(2) p2(2)];
dx=diff(x);
dy=diff(y);
hh(1)=plot(x,y,linespec);
ax=axis;
lx=diff(ax(1:2))*15e-3;
ly=diff(ax(3:4))*10e-3;
phi=atan2(dy,dx);
head=rotate2([-lx 0 -lx;ly 0 -ly],[0;0],phi);
col=get(hh(1),'color');
hh(2)=patch(head(1,:)+p2(1),head(2,:)+p2(2),col);
set(hh(2),'edgec',col)
set(hh(1),'user',{'vector',p1,p2,hh(2)})
set(hh(2),'user',{'vector',p1,p2,hh(1)})
if ~held,hold off,end
if nargout,h=hh;end
% ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

function newpoints=rotate2(points,origo,phi)
%ROTATE2 rotate points in 2 dimensions.
% NEWPOINTS=ROTATE(POINTS,ORIGO,PHI) rotates the
% points in POINTS PHI radians around the ORIGO
% point.
% ROTATE returns NEWPOINTS as the rotated points from POINTS.

% Copyright(c) 2000-05-14 by B. Rasmus Anthin.

A=[cos(phi) -sin(phi);sin(phi) cos(phi)];
newpoints=A*points+origo*ones(1,size(points,2));
% ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

== '''Examples''' ==
----
'''Example 1''': If a bar magnet is located at the origin with its North end aligned with the positive X-axis, what are the directions of the magnetic field at the following observation locations: above, below, to the left, to the right, and in a plane that is above the magnet?

We already know that the field of a bar magnet flows out of the north end and into the south end in a curling fashion. So, using the diagram above, it is easy to see that to the right of the magnet, the direction of the magnetic field points in the +X direction. At a position to the left of the magnet, the field is flowing back into the south end of the magnet, so the direction of the magnetic field at this location is also in the +X direction.

The field above and below the magnet is flowing from the right to the left at both locations, so the direction of the magnetic field above and below the magnet is in the -X direction.

At a different plane (z ≠ 0), there is no magnetic field, because we can assume that bar magnet acts as a 2-D dipole.

'''Example 2:''' A bar magnet with magnetic dipole moment 0.58 A∙m2 lies on the negative x axis, as shown in the figure below. A compass is located at the origin. Magnetic north is in the negative z direction. Between the bar magnet and the compass is a coil of wire of radius 3.5 cm, connected to batteries not shown. The distance from the center of the coil to the center of the compass is 9.6 cm. The distance from the center of the bar magnet to the center of the compass is 23.0 cm. A steady current of 0.96 A runs through the coil. Conventional current runs clockwise in the coil when viewed from the location of the compass. Despite the presence of the magnet and coil, the compass still points north.

'''(a)''' Which pole of the bar magnet is closer to the compass?
'''(b)''' How many turns of wire are in the coil?

'''Part A:''' Because the conventional current runs clockwise in the coil, you can use right hand rule to determine what direction the magnetic field is due to the coil. This tells us that the magnetic field due to the coil is in the -X direction. In order for the compass to stay still, the magnet needs to directly oppose the magnetic field of the coil, meaning its magnetic field has to point in the +X direction, meaning the '''north pole''' would have to be nearer the compass.

'''Part B:''' Because the magnetic field created by the coil is equal to the magnetic field created by the magnet, we can set their two fields equal to each other:
<math> \frac{\mu _{0}}{4 \pi } \cdot \frac{2\mu }{r^{3}} = \frac{\mu _{0}}{4 \pi } \cdot \frac{2NI\pi R^{2}}{(z^{2}+R^{2})^{3/2}} </math>

Rearranging to solve this equation for N, we get: <math> N = \frac{\mu (z^{2}+R^{2})^{3/2}}{I\pi R^{2} d^{3}} </math>

Plugging in .58 A∙m2 (the magnetic dipole moment, μ), .096 meters for z, .035 meters for R, .96 Amps for I, and .23 meters for d, we get that the number of loops in the coil is '''14'''.

== '''Connectedness''' ==
----
[[File:Series L0.JPG|thumb|left|An experimental MAGLEV train created by Japanese engineers.]]
One very interesting applications of magnets is their ability to levitate objects. This is the main driving force in the case of MAGLEV trains. Magnetic levitation, or MAGLEV trains, hover above a long series of magnets where the magnets on the bottom of the train repel the magnets on the tracks below it. Sending an electric current through the coils on the bottom of the track allows the train to levitate a few inches off the ground, and propelling the current through the guided coils on the bottom of the track propels the train forward at unbelievable speeds (up to 250 MPH)!

Making the train levitate is a useful tool because it reduces the amount of friction between the wheels and the track, and it allows for less fossil fuels to be used in order to make the train propel forwards.

[[File:img.jpg|thumb|left|MRI of brain.]]
Magnetism is also used in medical technology. Medical Resonance Imaging (MRI) machines use magnetic fields and radio waves to create images of the body.

== '''History''' ==
----
[[File:James Clerk Maxwell.png|thumb|right|James Clerk Maxwell]]
The first magnets were not invented, but rather discovered. The ancient Greeks and ancient Chinese stumbled upon a naturally occurring material, called magnetite, by mistake. People were so astounded by it that tales were told of magical islands where magnetic nature was everywhere. The Chinese actually developed a compass around 4500 years ago using this magnetite!

Despite not being the first people to study magnetism, Hans Christian Oersted did prove that electricity and magnetism were related by bringing a current carrying wire close to a compass needle. However, it wasn't until Maxwell published his findings in 1862 that led to the relationships between electricity and magnetism (Maxwell's Equations; see other Wikipedia page).

== '''External links''' ==
----
# MAGLEV Trains: http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-3-Biology-Vol-1/Magnetism-Real-life-applications.html
# More information on Bar Magnets: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html

== '''References''' ==
----
# http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
# https://en.wikipedia.org/wiki/Magnet#/media/File:VFPt_cylindrical_magnet_thumb.svg
# http://www.howmagnetswork.com/history.html
# https://en.wikipedia.org/wiki/Maglev#/media/File:Series_L0.JPG
# https://en.wikipedia.org/wiki/James_Clerk_Maxwell#/media/File:James_Clerk_Maxwell.png
Category: '''Fields'''

'''Created by: John Joyce'''

__FORCETOC__

Magnetic Field of a Toroid Using Ampere's Law

2016-11-28T00:16:06Z

Ycho78:

Claimed by Kevin McGorrey

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
 <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

 The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
 <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
 From this, we can solve for the magnetic field for a toroid:
 <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

 
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

 
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

 
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

 
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

 
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

== See also ==

[[Ampere's Law]]
 
[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
 
[[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===
*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html
*http://www.phys.uri.edu/gerhard/PHY204/tsl242.pdf
*https://clas-pages.uncc.edu/phys2102/online-lectures/chapter-7-magnetism/7-3-amperes-law/example-magnetic-field-of-a-toroid/

==References==
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. Pg 888-889.
 
Fall 2014 Test 4 from Phys 2212.
[[Category:Maxwell's Equations]]

Magnetic Field of a Toroid Using Ampere's Law

2016-11-28T00:13:34Z

Ycho78:

Claimed by Kevin McGorrey Claimed by Yeon Jae Cho (2016)

This page explains how to use Ampere's Law to solve for the magnetic field of a toroid.

==Magnetic Field of a Toroid using Ampere's Law==

Using Ampere's Law simplifies finding the magnetic field of a toroid since using the Biot-Savart law would be extremely difficult due to having to integrate over all the current elements in the toroid.
===Geometry of a Toroid===
[[File:Geometry Fig.JPG |frame|right|alt=Alt text| A toroid's geometry.]]
A toroid is essentially a solenoid whose ends meet. It is shaped like a doughnut (the base shape does not need to be a circle; it can be a triangle, square, quadrilateral, etc.), is symmetrical around an axis of rotation, and contains ''N'' loops around a closed, circular path with a radius of ''r'' inside of its loop.

===A Mathematical Model===
[[File:Computation Fig.JPG | left|frame|none|alt=Alt text| A toroid's view from above.]]
First we start with solving the path integral from Ampere's law:
 <div style="text-align: center;"><math>{\oint\,\vec{B}•d\vec{l} = μ_{0}∑I_{inside&ensp;path}}</math></div>

 The magnetic field, <math>{\vec{B}}</math>, is, due to the symmetry of a toroid, constant in magnitude and always tangential to the circular path (i.e. parallel to <math>{d\vec{l}}</math>). The path of a toroid is circular, so <math>{\oint\,d\vec{l}}</math> is equal to <math>{2πr}</math>. Therefore, the path integral of the magnetic field is equal to <math>{B2πr}</math>. The amount of current piercing the soap film (i.e. <math>{∑I_{inside&ensp;path}}</math>) is <math>NI</math>, where <math>N</math> is the number of piercings (i.e. turns in the coil) and <math>I</math> is the current. Ampere's law is now this:
 <div style="text-align: center;"><math>{B2πr = μ_{0}NI}</math></div>
 From this, we can solve for the magnetic field for a toroid:
 <div style="text-align: center;"><math>{B = \frac{μ_{0}NI}{2πr}}</math></div>

 
==Examples==
===Simple===
[[File:Simple Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the simple example.]]
A toroid frame is made out of plastic of small square cross section and tightly wrapped uniformly with 100 turns of wire, so that the magnetic field has essentially the same magnitude throughout the plastic (radius R of the curved part is much larger than cross section width w). With a current of 2 A and radius of 5 m, what is the magnetic field inside the plastic.
[[File:Simple Example Sol.JPG | center|frame|none|alt=Alt text| The solution for the simple example.]]

===Middling - Difficult===
[[File:Mid - Diff Example Fig.JPG | right|frame|none|alt=Alt text| Figure for the middle and difficult examples.]]
The toroid shown in the diagram has an inner radius of <math>R_{i}</math> and an outer radius of <math>R_{o}</math> and is centered at the origin in the diagram. The z-axis passes through the center of the doughnut hole. This toroid is wrapped with <math>N</math> loops of current <math>I</math> flowing up the outside surface of the toroid, radially inward, down the inner surface, and then radial outward. Assume that the magnetic field produced by this toroid has the form <math>\vec{B} = B(r,z)\hat{φ}</math> '''at every point in space''' where <math>r</math> is the perpendicular distance from the z-axis and <math>\hat{φ}</math> is a unit vector which "curls" around the z-axis, i.e., it is always tangent to any circle with rotational symmetry around the z-axis.

 
====Middling====
(a.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r < R_{i}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

 
(b.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>r > R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.
[[File:Mid - Diff Example Sol ab.JPG | center|frame|none|alt=Alt text| The solution for the middle example.]]

 
====Difficult====
(c.) Consider a z-axis centered Amperian loop in the plane of the toroid, at <math>z = 0</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field inside the inner radius of the toroid.

 
(d.) Consider a z-axis centered Amperian loop far above the toroid <math>z >> R_{o}</math>, with a radius <math>R_{i} < r < R_{o}</math> and use it to find the magnitude of the magnetic field far above the toroid.
[[File:Mid - Diff Example Sol cd.JPG | center|frame|none|alt=Alt text| The solution for the difficult example.]]

== See also ==

[[Ampere's Law]]
 
[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
 
[[Magnetic Field of Coaxial Cable Using Ampere's Law]]

===Further reading===
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley.

===External links===
*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html
*http://www.phys.uri.edu/gerhard/PHY204/tsl242.pdf
*https://clas-pages.uncc.edu/phys2102/online-lectures/chapter-7-magnetism/7-3-amperes-law/example-magnetic-field-of-a-toroid/

==References==
Chabay, Sherwood. (2015). Matter and Interactions (4th ed., Vol. 2). Raleigh, North Carolina: Wiley. Pg 888-889.
 
Fall 2014 Test 4 from Phys 2212.
[[Category:Maxwell's Equations]]

Ampere-Maxwell Law

2016-11-27T23:39:53Z

Ycho78:

Claimed by Maria Rivero
Claimed by Yeon Jae Cho (FALL 2016)

== Ampere-Maxwell Law==

Ampere-Maxwell Law was discovered in the 1800's by James Clerk Maxwell. Maxwell proved in his paper that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

Ampere-Maxwell Law is also known as the Ampere's Circuital Law, and it shall not be confused with Ampere's Force Law.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

From the picture we see that the speed of light relates the a time varying electric and magnetic field.

==What this implies==

In the example above we saw that the speed of light relates the a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

==Conceptual Question==
[[File: AMconceptual.png|500px|Image: 500 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

[[File: AMcapacitor.png|500px|Image: 500 pixels]]

Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is

1) Zero

2) Positive

3) Negative

4) Can’t tell

Answer:

One thing to notice is that there is no enclosed current through the disk. When integrating in the direction shown, the electric flux is positive. Because the plates are charging, the electric flux is increasing. Therefore the line integral is positive.

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

I am particularly interested in understanding medicine and how using all these waves we can help doctors understand and see different aspects of the human body.

How is it connected to your major?

I do not know how this would be connected to Industrial Engineering,
however it is very useful for Electrical Engineering and Mechanical Engineering.

Is there an interesting industrial application?

As I mentioned before, this is very useful in medicine when creating the images for the
MRI scanners in hospitals. But it is also important to generate electricity, to build computers
and phones, etc. More generally, Maxwell's equations work for all devices that use electricity and magnets.

== See also ==

Maxwell's Equations:
http://study.com/academy/lesson/maxwells-equations-definition-application.html

James Maxwell Biography:
http://www.biography.com/people/james-c-maxwell-9403463

==References==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

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

Ampere-Maxwell Law

2016-11-27T23:37:20Z

Ycho78:

Claimed by Maria Rivero
Claimed by Yeon Jae Cho (FALL 2016)

== Ampere-Maxwell Law==

Ampere-Maxwell Law was discovered in 1861 by James Clerk Maxwell in his paper "On Physical Lines of Force". Maxwell proved in this paper that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

From the picture we see that the speed of light relates the a time varying electric and magnetic field.

==What this implies==

In the example above we saw that the speed of light relates the a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

==Conceptual Question==
[[File: AMconceptual.png|500px|Image: 500 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

[[File: AMcapacitor.png|500px|Image: 500 pixels]]

Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is

1) Zero

2) Positive

3) Negative

4) Can’t tell

Answer:

One thing to notice is that there is no enclosed current through the disk. When integrating in the direction shown, the electric flux is positive. Because the plates are charging, the electric flux is increasing. Therefore the line integral is positive.

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

I am particularly interested in understanding medicine and how using all these waves we can help doctors understand and see different aspects of the human body.

How is it connected to your major?

I do not know how this would be connected to Industrial Engineering,
however it is very useful for Electrical Engineering and Mechanical Engineering.

Is there an interesting industrial application?

As I mentioned before, this is very useful in medicine when creating the images for the
MRI scanners in hospitals. But it is also important to generate electricity, to build computers
and phones, etc. More generally, Maxwell's equations work for all devices that use electricity and magnets.

== See also ==

Maxwell's Equations:
http://study.com/academy/lesson/maxwells-equations-definition-application.html

James Maxwell Biography:
http://www.biography.com/people/james-c-maxwell-9403463

==References==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

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

Ampere-Maxwell Law

2016-11-27T23:36:57Z

Ycho78:

Claimed by Maria Rivero

== Ampere-Maxwell Law==

Ampere-Maxwell Law was discovered in 1861 by James Clerk Maxwell in his paper "On Physical Lines of Force". Maxwell proved in this paper that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

[[File: Ampere-Maxwell.png]]

Where
B is the magnetic field
dl is the change in path
The sum of I is the sum of the charges inside the path.

==Example==

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

From the picture we see that the speed of light relates the a time varying electric and magnetic field.

==What this implies==

In the example above we saw that the speed of light relates the a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

==Conceptual Question==
[[File: AMconceptual.png|500px|Image: 500 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

[[File: AMcapacitor.png|500px|Image: 500 pixels]]

Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is

1) Zero

2) Positive

3) Negative

4) Can’t tell

Answer:

One thing to notice is that there is no enclosed current through the disk. When integrating in the direction shown, the electric flux is positive. Because the plates are charging, the electric flux is increasing. Therefore the line integral is positive.

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

I am particularly interested in understanding medicine and how using all these waves we can help doctors understand and see different aspects of the human body.

How is it connected to your major?

I do not know how this would be connected to Industrial Engineering,
however it is very useful for Electrical Engineering and Mechanical Engineering.

Is there an interesting industrial application?

As I mentioned before, this is very useful in medicine when creating the images for the
MRI scanners in hospitals. But it is also important to generate electricity, to build computers
and phones, etc. More generally, Maxwell's equations work for all devices that use electricity and magnets.

== See also ==

Maxwell's Equations:
http://study.com/academy/lesson/maxwells-equations-definition-application.html

James Maxwell Biography:
http://www.biography.com/people/james-c-maxwell-9403463

==References==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

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