Physics Book - User contributions [en]

Magnetic Flux

2017-04-09T15:23:38Z

Rahulsingi:

'''claimed by Rahul Singi (Spring 2017)'''

==The Main Idea==
===A Mathematical Model===
Recall that according to Gauss' law, the electric flux through any closed surface is directly proportional to the net electric charge enclosed by that surface. Given the very direct analogy which exists between an electric charge and a magnetic 'monopoles', we would expect to be able to formulate a second law which states that the magnetic flux through any closed surface is directly proportional to the number of magnetic monopoles enclosed by that surface. But the problem is that magnetic monopoles don't exist. It follows that the equivalent of Gauss' law for magnetic fields reduces to:

<math>\Phi_B = \oint B \cdot dA = 0</math>

Realistically, the magnetic flux though any CLOSED surface is zero. The magnetic flux through an area will be its own individual value. This rule is useful when solving for a an unknown magnetic field that's coming from a side of a surface when the other fields from the other sides are known. Magnetic flux is measured in SI units of Weber (WB), named after German physicist and co-inventor of the telegraph Wilhelm Weber.

===A Computational Model===
As seen on the formula described on the section above, the magnetic flux of an object depends on the size and shape of the object. Therefore, it is hard to show one specific computational model that can correctly illustrate magnetic flux. However, below are some images that show the magnetic flux of three common types of surfaces: closed, open, and solenoid.

A closed surface: [[File:closedsingi.gif]]

An open surface: [[File:opensingi.gif]]

Solenoid: [[File:solenoidsingi.gif]]

==Further Description==

Gauss's Law for magnetism tells us that magnetic monopoles do not exist. If magnetic monopoles existed, they would be sources and sinks of the magnetic field, and therefore the right-hand side could differ from zero. Gauss's Law for magnetism is one of the four Maxwell's equations, which form the foundation for the entire theory of classical electrodynamics.

The magnitude of the magnetic flux depends on the strength of the magnetic field, the size of the surface area, and the angle between the direction in which the surface area points and the direction of the magnetic field.

[[File:MagFlux.gif]]

==Practice Problems==
===Walkthroughs===

Simple Walthrough:[https://www.youtube.com/watch?v=xKEEOEvYVB4]

Middling Walkthrough: [https://www.youtube.com/watch?v=M0SRan7UW6k]

Advanced Walkthrough: [https://www.youtube.com/watch?v=ba8ADUeGd9w]

Once you get the hang of the process of solving magnetic flux problems using the walkthroughs, then move on to the more advanced questions below.
===Advanced===

1) There is a small bar magnet with a magnetic dipole D located at the origin (0,0,0). It's aligned with the y-axis. There is a circular disk with a radius of R facing perpendicular to the yz plane and its center is 4 meters away on the +x axis from the bar magnet. What is the magnetic flux going through the disk in terms of the given variable? (Consult with your professor for the solution).

2) Referring back to the previous problem, the disk is now tilted so that the angle between the yz plane and the surface is 30 degrees. Find the new magnetic flux in terms of the given variables.

3) A square with side length T is directly facing the xy plane 3 meters away from a current carrying 1 meter wire (from a portion of a nearby circuit powered by a battery with an emf of U). The wire is aligned with the y axis. The magnetic flux going through the square is G. Find the resistance of the wire.

==Connectedness==
Magnetic flux is very closely related to electric flux using Gauss's law for electric fields (one of the four Maxwell equations).

In the real world, magnetic flux is used for several different reasons. First, the use of magnetic flux is very important when looking at closed surfaces, because the flux is always zero. This is allows us to simplfy complex magnetic field calculations involving those type of surfaces. Additionally, magnetic flux and its laws allows us to calculate the voltage generated by an electric generator even when the magnetic field is complicated, easily.

==History==
A major part of the history of magnetic flux is tied to Carl Freidrich Gauss. Gauss was a 19th century German physicist and mathematician that had notable contributions in not only the world of physics but across the fields of algebra, astronomy, and geometry. However, in this course, he is mainly known for the introduction of one of the four Maxwell's Laws(partial differential equations), titled Gauss's Law which describes the equation used to calculate magnetic flux.

Prior to Gauss's study on the subject however, Joseph-Louis Lagrange (famous Italian mathematician) had studied the subject. However, Gauss completed the formulation of the theory and published it in 1813.

Lagrange: [[File:lgsingi.jpg]]

Gauss: [[File:gsingi.jpg]]

==Further Reading==
What is Magnetic Flux?: [https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-flux-faradays-law/a/what-is-magnetic-flux]

Overview: [http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html]

In-Depth Tutorial: [http://www.electronics-tutorials.ws/electromagnetism/magnetism.html]

==Helpful Links/Videos==
Khan Academy: [https://www.khanacademy.org/science/physics/magnetic-forces-and-magnetic-fields/magnetic-flux-faradays-law/v/flux-and-magnetic-flux]

Quick Explanation: [https://www.youtube.com/watch?v=3NYg34_vy7k]

In-Depth Explanation: [https://www.youtube.com/watch?v=1RcpSUr8GtA]

==References==
Pictures: [http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html]

Pictures: [http://wiki.kidzsearch.com/wiki/Magnetic_flux]

Pictures: [http://physics.stackexchange.com/questions/188503/how-do-magnetic-field-lines-cancel-outside-of-a-solenoid]

Pictures: [http://tempuss.it/2016/01/25/la-matematica-di-lagrange/g-l-lagrange/]

Pictures: [https://www.rare-earth-magnets.com/johann-carl-friedrich-gauss]

Magnetic Flux

2017-04-08T19:04:11Z

Rahulsingi:

Magnetic Flux

2017-04-08T18:17:09Z

Rahulsingi:

'''claimed by Rahul Singi (Spring 2017)'''

==The Main Idea==
===A Mathematical Model===
Recall that according to Gauss' law, the electric flux through any closed surface is directly proportional to the net electric charge enclosed by that surface. Given the very direct analogy which exists between an electric charge and a magnetic 'monopoles', we would expect to be able to formulate a second law which states that the magnetic flux through any closed surface is directly proportional to the number of magnetic monopoles enclosed by that surface. But the problem is that magnetic monopoles don't exist. It follows that the equivalent of Gauss' law for magnetic fields reduces to:

<math>\Phi_B = \oint B \cdot dA = 0</math>

Realistically, the magnetic flux though any CLOSED surface is zero. The magnetic flux through an area will be its own individual value. This rule is useful when solving for a an unknown magnetic field that's coming from a side of a surface when the other fields from the other sides are known.

===A Computational Model===
As seen on the formula described on the section above, the magnetic flux of an object depends on the size and shape of the object. Therefore, it is hard to show one specific computational model that can correctly illustrate magnetic flux. However, below are some images that show the magnetic flux of three common types of surfaces: closed, open, and solenoid.

A closed surface: [[File:closedsingi.gif]]

An open surface: [[File:opensingi.gif]]

Solenoid: [[File:solenoidsingi.gif]]

==Further Description==

Gauss's Law for magnetism tells us that magnetic monopoles do not exist. If magnetic monopoles existed, they would be sources and sinks of the magnetic field, and therefore the right-hand side could differ from zero. Gauss's Law for magnetism is one of the four Maxwell's equations, which form the foundation for the entire theory of classical electrodynamics.

The magnitude of the magnetic flux depends on the strength of the magnetic field, the size of the surface area, and the angle between the direction in which the surface area points and the direction of the magnetic field.

[[File:MagFlux.gif]]

==Practice Problems==
===Walkthroughs===

Simple Walthrough:[https://www.youtube.com/watch?v=xKEEOEvYVB4]

Middling Walkthrough: [https://www.youtube.com/watch?v=M0SRan7UW6k]

Advanced Walkthrough: [https://www.youtube.com/watch?v=ba8ADUeGd9w]

Once you get the hang of the process of solving magnetic flux problems using the walkthroughs, then move on to the more advanced questions below.
===Advanced===

1) There is a small bar magnet with a magnetic dipole D located at the origin (0,0,0). It's aligned with the y-axis. There is a circular disk with a radius of R facing perpendicular to the yz plane and its center is 4 meters away on the +x axis from the bar magnet. What is the magnetic flux going through the disk in terms of the given variable? (Consult with your professor for the solution).

2) Referring back to the previous problem, the disk is now tilted so that the angle between the yz plane and the surface is 30 degrees. Find the new magnetic flux in terms of the given variables.

3) A square with side length T is directly facing the xy plane 3 meters away from a current carrying 1 meter wire (from a portion of a nearby circuit powered by a battery with an emf of U). The wire is aligned with the y axis. The magnetic flux going through the square is G. Find the resistance of the wire.

==Connectedness==
Magnetic flux is very closely related to electric flux using Gauss's law for electric fields (one of the four Maxwell equations).

In the real world, magnetic flux is used for several different reasons.

==History==
A major part of the history of magnetic flux is tied to Carl Freidrich Gauss. Gauss was a 19th century German physicist and mathematician that had notable contributions in not only the world of physics but across the fields of algebra, astronomy, and geometry. However, in this course, he is mainly known for the introduction of one of the four Maxwell's Laws(partial differential equations), titled Gauss's Law which describes the equation used to calculate magnetic flux.

Prior to Gauss's study on the subject however, Joseph-Louis Lagrange (famous Italian mathematician) had studied the subject. However, Gauss completed the formulation of the theory and published it in 1813.

Lagrange: [[File:lgsingi.jpg]]

Gauss: [[File:gsingi.jpg]]

File:Lgsingi.jpg

2017-04-08T18:16:22Z

Rahulsingi:

File:Gsingi.jpg

2017-04-08T18:15:03Z

Rahulsingi:

Magnetic Flux

2017-04-08T17:16:39Z

Rahulsingi:

Magnetic Flux

2017-04-08T17:13:48Z

Rahulsingi:

File:Solenoidsingi.gif

2017-04-08T17:13:06Z

Rahulsingi:

File:Opensingi.gif

2017-04-08T17:12:36Z

Rahulsingi:

Magnetic Flux

2017-04-08T17:12:04Z

Rahulsingi:

File:Closedsingi.gif

2017-04-08T17:11:12Z

Rahulsingi:

Magnetic Flux

2017-04-08T17:09:28Z

Rahulsingi:

Magnetic Flux

2017-04-08T16:58:56Z

Rahulsingi:

Magnetic Flux

2017-04-08T16:47:55Z

Rahulsingi:

Magnetic Flux

2017-04-08T16:46:50Z

Rahulsingi:

Magnetic Flux

2017-04-08T16:45:12Z

Rahulsingi:

Magnetic Flux

2017-04-08T16:36:50Z

Rahulsingi:

Bar Magnet

2017-04-04T23:54:31Z

Rahulsingi:

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__

Magnetic Flux

2017-04-04T23:53:15Z

Rahulsingi:

'''claimed by Rahul Singi (Spring 2017)'''

==The Main Idea==
Recall that according to Gauss' law, the electric flux through any closed surface is directly proportional to the net electric charge enclosed by that surface. Given the very direct analogy which exists between an electric charge and a magnetic 'monopoles', we would expect to be able to formulate a second law which states that the magnetic flux through any closed surface is directly proportional to the number of magnetic monopoles enclosed by that surface. But the problem is that magnetic monopoles don't exist. It follows that the equivalent of Gauss' law for magnetic fields reduces to:

<math>\Phi_B = \oint B \cdot dA = 0</math>

Realistically, the magnetic flux though any CLOSED surface is zero. The magnetic flux through an area will be its own individual value. This rule is useful when solving for a an unknown magnetic field that's coming from a side of a surface when the other fields from the other sides are known.

==Further Description==

Gauss's Law for magnetism tells us that magnetic monopoles do not exist. If magnetic monopoles existed, they would be sources and sinks of the magnetic field, and therefore the right-hand side could differ from zero. Gauss's Law for magnetism is one of the four Maxwell's equations, which form the foundation for the entire theory of classical electrodynamics.

The magnitude of the magnetic flux depends on the strength of the magnetic field, the size of the surface area, and the angle between the direction in which the surface area points and the direction of the magnetic field.

[[File:MagFlux.gif]]

==Practice Problems==

1) There is a small bar magnet with a magnetic dipole D located at the origin (0,0,0). It's aligned with the y-axis. There is a circular disk with a radius of R facing perpendicular to the yz plane and its center is 4 meters away on the +x axis from the bar magnet. What is the magnetic flux going through the disk in terms of the given variable? (Consult with your professor for the solution).

2) Referring back to the previous problem, the disk is now tilted so that the angle between the yz plane and the surface is 30 degrees. Find the new magnetic flux in terms of the given variables.

3) A square with side length T is directly facing the xy plane 3 meters away from a current carrying 1 meter wire (from a portion of a nearby circuit powered by a battery with an emf of U). The wire is aligned with the y axis. The magnetic flux going through the square is G. Find the resistance of the wire.

Bar Magnet

2017-04-04T23:42:58Z

Rahulsingi:

'''claimed by Rahul Singi (Spring 2017)'''

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''' ==
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[[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''' ==
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# 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''' ==
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# 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'''

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