Physics Book - User contributions [en]

Magnetic Dipole

2025-12-01T17:58:28Z

Zeynepuzun: /* Magnetic Flux and Degaussing Coils */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:Magdipim2.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:Dipolepattern.png]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:Currentloopfixed.png]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]
[https://msi.nga.mil/api/publications/download?key=16920950/SFH00000/HoMCA.pdf&type=view| Source]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:58:19Z

Zeynepuzun: /* Magnetic Flux and Degaussing Coils */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:Magdipim2.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:Dipolepattern.png]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:Currentloopfixed.png]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]
[https://msi.nga.mil/api/publications/download?key=16920950/SFH00000/HoMCA.pdf&type=view| Source]
This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:57:14Z

Zeynepuzun: /* A Mathematical Model */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:Magdipim2.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:Dipolepattern.png]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:Currentloopfixed.png]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Currentloopfixed.png

2025-12-01T17:57:04Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T17:52:21Z

Zeynepuzun: /* Connection With Magnetic Fields in Loops of Wire */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:Magdipim2.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:Dipolepattern.png]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Dipolepattern.png

2025-12-01T17:52:11Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T17:49:19Z

Zeynepuzun: /* Direction of Dipole Moment */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:Magdipim2.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:48:45Z

Zeynepuzun: /* Connection With Magnetic Fields in Loops of Wire */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:Magdipim2.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Magdipim2.png

2025-12-01T17:48:30Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T17:44:20Z

Zeynepuzun: /* The Main Idea */

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magdipoledraw.png]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Magdipoledraw.png

2025-12-01T17:44:09Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T17:38:49Z

Zeynepuzun:

Claimed by '''Zeynep Uzun (Fall 2025)'''

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:38:04Z

Zeynepuzun: /* Middling */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

;Example 2
: A bar magnet is placed a distance 15 cm to the right of a compass that originally points North. After the magnet is placed next to the compass, the needle of the compass deflects 45 degrees to the west. What is the magnitude of the magnetic moment of the bar magnet? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

'''Example 2 Solution'''

Follow a similar process as described above. The solution is <math> \mu = 0.675 A*M^2</math>

</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:26:53Z

Zeynepuzun: /* Computational Model */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

Click [https://www.glowscript.org/#/user/Zeynep_Uzun/folder/MyPrograms/program/magdipole| here] for a glowscript model visualizing the magnetic field of a bar magnet in 2D. You may click and drag the blue magnet to see how the field changes.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:21:05Z

Zeynepuzun: /* Computational Model */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

<iframe src="https://trinket.io/embed/glowscript/61ba188df3ef?start=result" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:19:39Z

Zeynepuzun: /* Computational Model */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

<iframe src="https://trinket.io/embed/glowscript/31d0f9ad9e?start=result" width="100%" height="356" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T17:01:52Z

Zeynepuzun: /* Simple */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:Ex2simplemagdip.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Ex2simplemagdip.png

2025-12-01T17:01:40Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T17:00:41Z

Zeynepuzun: /* Examples */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: Two identical square loops of wire have resistance R and sides of length d. The centers of the loop are equidistant from the origin at a distance L >> d. Loop 1 has a counterclockwise current flowing through it that is decreasing with time. What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1magdipdiff.png]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
First, identify the direction of the induced current on loop 2.
A loop of current from far away looks like a magnetic dipole. So, at loop 2, the magnetic field from loop 1 is into the page. The current in loop 1 is decreasing, so the flux into the page at loop 2 is decreasing. When flux is decreasing, nature counteracts it by inducing a flux into the page stronger, which requires a clockwise current. Thus, I2 is clockwise. The magnetic field of loop 1 is of that of a dipole on the perpendicular axis. The magnetic dipole moment is current multiplied by area; use this in the equation.

Second, you need to find the flux. The magnetic field is into the page and n-hat is out of the page. This means the dot product result will be negative.
Here is the full solution:
[[File:Ex2difficultmagdipsol.jpeg]]

</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Ex1magdipdiff.png

2025-12-01T16:59:45Z

Zeynepuzun:

File:Ex2difficultmagdipsol.jpeg

2025-12-01T16:57:55Z

Zeynepuzun:

File:Ex2difficultmagdip.jpeg

2025-12-01T16:42:11Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T16:34:41Z

Zeynepuzun: /* Difficult */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[[File:Ex1difficultmagdipole.jpeg]]

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T16:34:09Z

Zeynepuzun: /* Difficult */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.
[File:Ex1difficultmagdipole.jpeg]

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Ex1difficultmagdipole.jpeg

2025-12-01T16:33:39Z

Zeynepuzun:

Magnetic Dipole

2025-12-01T16:30:20Z

Zeynepuzun: /* Examples */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Difficult</u>===
;'''Example 1'''
: What is the magnetic flux on the right loop? Define +n-hat as +z-hat. You need to have an understanding of [https://www.physicsbook.gatech.edu/Magnetic_Flux magnetic flux] for this problem.

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T16:20:48Z

Zeynepuzun: /* Simple */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T16:19:38Z

Zeynepuzun: /* Simple */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg|5px|]]
[[File:ex2magdipole.jpeg|caption]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T16:19:22Z

Zeynepuzun: /* Simple */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg|50px|]]
[[File:ex2magdipole.jpeg|caption]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T16:18:54Z

Zeynepuzun: /* Simple */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg|200px|thumb|Caption]]
[[File:ex2magdipole.jpeg|caption]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T16:16:59Z

Zeynepuzun: /* Simple */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

;'''Example 2'''
: What is the direction of the magnetic field at the observation location?
[[File:ex2magdipole.jpeg|caption]]

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

'''Example 2 Solution'''
First, identify the direction of the magnetic dipole moment. The magnetic dipole moment points from the S pole to the N pole, so mu is pointing in the positive y-hat direction. Because the observation axis is perpendicular, the magnetic field will point opposite to the mu direction. Thus, the magnetic field at the observation location points in the negative y-hat direction.

</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

File:Ex2magdipole.jpeg

2025-12-01T16:12:41Z

Zeynepuzun:

File:JPEG image-4446-BB51-0E-0.jpeg

2025-12-01T16:08:08Z

Zeynepuzun: Simple Question 2 Magnetic Dipole Moment Image

== Summary ==
Simple Question 2 Magnetic Dipole Moment Image

Magnetic Dipole

2025-12-01T15:56:39Z

Zeynepuzun: /* Examples */

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===<u>Simple</u>===

;'''Example 1'''
: In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''
We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>
</div>
</div>

===<u>Middling</u>===
;'''Example 1'''
: When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
'''Example 1 Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>
</div>
</div>

===<u>Conceptual Questions</u>===
# If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]
<div class="mw-collapsible mw-collapsed">
'''''Click Expand for Solution'''''
<div class="mw-collapsible-content">
The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.
</div>
</div>

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-12-01T15:40:44Z

Zeynepuzun:

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===
==== ''A brief derivation:'' ====

From the Biot-Savart Law, we know that the [https://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop| magnetic field of a loop], when calculating the magnitude of the magnetic field at a point on the z-axis, is:
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{(z^2 + R^2)^{3/2}} \text{ ,where R is the radius of the circular loop, and z is the distance from the center of the loop} </math>
An approximation can be made if the distance from the center of the loop is much greater than the radius of the loop (z >> R):
::<math>B = \frac{\mu_0}{4 \pi} \frac{2 \pi I R^2}{z^3} \text{ ,where z is much greater than R} </math>

Given the magnetic dipole moment is current (I) multiplied by area (pi R^2), we can derive the magnetic dipole on axis as:
::<math>B = \frac{\mu_0}{4\pi} \frac{2\mu}{z^3}</math>

----

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===Simple===

In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

'''Solution'''

We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

===Middling===

When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

'''Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

===Conceptual Question===

If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]

'''Solution'''

The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Magnetic Dipole

2025-11-30T21:27:18Z

Zeynepuzun:

Claimed by Zeynep Uzun (Fall 2025)

==The Main Idea==

A dipole is a pair of field producing entities placed close together to produce a particular pattern of field. In such cases it is useful to look a the pair as a single entity as a simplification as it allows us to create simple and commonly applicable equations for them as opposed to considering the two entities as separate. Dipoles also have a number of properties that shape the natural world in unique ways as the interactions of dipoles often create emergent properties.

Physics students are typically taught about electric dipoles first, and will recognize that many of the properties and formulas resemble each other quite closely, This is because dipoles have the same field shape no matter what field they are influencing and so the same relationships between position and intensity exist with respect to the magnitudes of the charges.

A common example of a macroscopic magnetic dipole is a common permanent magnet such as a bar magnet. These magnets will produce a magnetic field everywhere in space and the force varies only with observation position. This is very similar to an electric dipole, but instead of electrical charges, we have magnetic ones, represented by the North and South poles. Just like with positively and negatively charged ions, like charges repel, and opposite charges attract.

[[File:Magnetic_dipole.jpg]]

Since magnets always have two poles, the magnetic field produced is that of a magnetic dipole. The North pole is analogous to a positive charge and South to a negative charge, in that the field lines flow out from the North pole into the South pole.

[[File:Magnetificdipole1.jpg]]

===Major Distinction from Electrical Dipoles===

One major distinction between electrical and magnetic dipoles is the fact that while electrical "monopoles" exist in abundance in electrical charges, a magnetic monopole has never been confirmed to exist. This means that when talking about magnetism, we are essentially always talking in terms of dipoles and the dipole equations are used much more commonly than the formulas for individual magnetic charges. By using the formulas for dipoles, we can figure out the dipole moment of any magnet as long as we know our relative position and then we will have a reasonable model of any magnet that we can measure.

Bar magnets are what we typically think about when talking about dipoles, but there are a couple of other important constructs that create dipole magnetic fields. The other most notable magnetic dipole is the coil of wire with current flowing through it.

===Connection With Magnetic Fields in Loops of Wire===
One of the most important things to recognize is the similarity between an obvious dipole like a bar magnet and what occurs when current flows through a loop of wire and how that allows us to perform calculations much easier. Current moving in a closed loop will create a dipole magnetic field as shown below.

[[File:WireLoopDipole.png]]

We can see how this is similar to the field produced by particles(or equivalently by a permanent magnet)

[[File:DipoleField.jpg]]

One of the most important things to note about the dipole simplification is that it grants us the ability to calculate dipole moment, a measure of the polarity of the dipole, which aids in calculations regarding the energy and forces involved in iterating with the dipole. This is especially useful for dealing with current in a loop of wire, as it does not resemble a traditional dipole. Once you have the moment you can use all of the nice dipole formulas instead of trying to treat it as coils of wire.

===Magnetic Dipole Moment and Connection with Torque on a Coil===

The magnetic dipole moment represents the magnitude of the field of the dipole, essentially its magnetic potential.

The most common unit for the dipole moment is Ampere-square meters which is derived from how you calculate the moment of a coil of wire. It can also have the units of Joules per Tesla, representing the amount of energy per unit of magnetism an object will experience in the field of the dipole.

The magnetic dipole moment as it relates to current in coils of wire represents the torque on the current in the wire, which again is proportional to the magnitude of the magnetic field.

===A Mathematical Model===

The equation for a [http://www.physicsbook.gatech.edu/Magnetic_Dipole_Moment| magnetic dipole moment]
is:

<math> \boldsymbol{\mu} = \boldsymbol{IA} = \boldsymbol{I{\pi}R^2}</math> with units <math> A*m^2 </math>

where I is the current, and A is the cross sectional area. This will be used to calculate the magnitude of the magnetic field produced by a magnetic dipole.

The 2nd part of the equation is specifically for a circular loop field induced magnetic dipole and its area is the area of the loop.

From this equation, we can deduce the magnetic dipole moments just knowing the conventional current flowing through the loop and the radius.

[[File:current_loop.JPG]]

However, most of the time the current is not given. Furthermore, the equation is not applicable for the normal [http://www.physicsbook.gatech.edu/Bar_Magnet| bar magnets], for they do not have a electrical current flowing through. Thus, another way to get the dipole moment is by using the relationship between the magnetic dipole and the magnetic field induced by the dipole.

There are two equations based on the observation location.

==== Observation Location Perpendicular to Dipole Axis ====

If the observation location is perpendicular to the axis of the dipole(see below),

[[File:dipole_perp.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:Magpar.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is because magnetic dipoles and electric dipoles both have the same field shape.

==== Observation Location Along Dipole Axis ====

If the observation location is placed along the axis like in the image below,

[[File:dipole_along.JPG]]

then the equation for the magnetic field induced by the dipole is:

[[File:magper.JPG]]

Note that this equation is analogous to the equation for an observation location perpendicular to the axis of an electric dipole. This is again because magnetic dipoles and electric dipoles both have the same field shape.

==== Direction of Dipole Moment ====

It is also important to note the direction of the dipole moment. In a bar magnet the dipole moment will be wrapping from the North end of the magnet until it faces the South end of the magnet in a traditional dipole fashion and field pattern.

[[File:Magdidirection.gif]]

On the other hand, if you are looking at the dipole moment induced by a current-flowing loop, you have to use the right hand rule.
If you point your thumb in the direction of the current along the wire, your fingers will curl in the direction through the loop in the direction the magnetic field is flowing. Alternatively you can think about it another way and curl your fingers in the direction of the current, and your thumb will point in the direction of the magnetic dipole moment induced from the current.

===Computational Model===

The computational plot of a magnetic dipole is very similar to the plot of an electric dipole because of their similar forces. In some simulations, you can see a loop or coil of wire with the surrounding forces displayed using spaced vectors. In [https://www.compadre.org/osp/items/detail.cfm?ID=12361 this] simulation, you are able to move a dipole around and see in 3D how the dipole affects its surroundings with a constant magnetic field.

==Examples==

===Simple===

In a circular loop with current of 3 Ampere and diameter of 16 cm, what is the magnetic dipole moment induced from the current?

'''Solution'''

We can just simply use the first equation given in the beginning:

[[File:Magneticdipole1.JPG]]

where R is 0.08 meters and I is 3 Amperes. Calculating for the magnetic dipole moment gives:

<math> 3 \times 0.08^2 \times \pi = 0.0192 A*M^2</math>

===Middling===

When a compass is placed 20 cm away from a bar magnet parallel to the x-axis and shows deflection of 20 degrees, what is the magnitude of the magnetic dipole moment? You can assume that |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>

'''Solution'''

First, we need to figure out the magnetic dipole induced from the magnet. Thus, we need to use the formula:

[[File:Bearth.JPG]]

where |''B<sub>earth</sub>''| is <math>2 \times 10^-5 T</math>, and the deflection is 20 degrees. Thus,

<math> (2E-5 T) * tan(20) = 7.3E-6 T</math>

This gives us the magnitude of the magnetic field contributed by the bar magnet. We can plug in the value into the formula given (note that because it is along the x-axis, we use this formula)

[[File:Magper.JPG]]

<math> 7.3E-6 = \frac {1E-7 * 2 * \mu} {(0.2)^3} </math>

<math> \mu = 0.292 A*M^2</math>

===Conceptual Question===

If released from rest, what, if anything, will the current loop do?

[[File:Current_loop_mag_field.JPG]]

'''Solution'''

The loop will rotate counterclockwise. We know this because the magnetic dipole moment has a tendency to want to align itself with any externally applied magnetic field, if one exists. Using the right-hand rule for a loop of current, we curl our fingers clockwise, causing our thumb to point straight downwards. This means that our magnetic dipole moment is poiting straight downwards. In order to align itself with the magnetic field that is pointing to the right, it must rotate counterclockwise.

==Connectedness==

===Magnetic Flux and Degaussing Coils===

As someone in the Navy ROTC Unit and joining the fleet someday, I like to learn about navigation and am currently enrolled in naval navigation. In this class we have looked at many different things about maneuvering but the most interesting was about open ocean navigation and use of the [[#Magnetic North vs Geographic North|compass]](link to section on geographic vs. magnetic north). In the times before gyro compasses and GPS calculations to find true north, sailors had to purely rely on their magnetic compass.

[[File:Degaussing_coils.jpg]]

This was especially difficult due to the surrounding magnetic fields being emitted from the plethora of wires and equipment surrounding the compass on the bride and throughout the ship. So, in order to dampen the deviation of the compass created by the many different electric fields, the navy instituted what is called Degaussing Coils and spheres. These coils were strategically lined through the ship and the spheres were positioned directly next to the compass on opposite sides of the compass in order to cancel the surrounding magnetic field. These coils when used properly create a magnetic dipole.

In some robotics applications, like robotic soccer in my case, we use solenoids to rapidly accelerate iron cores to kick soccer balls at high velocities. The precise calculations to figure out how much acceleration the core is under and how the magnetic field affects it are immensely complex and require a lot of compute power and time, but we can use the dipole moment calculation to get a decent idea of how much energy is being imparted and a rough idea of the acceleration we are dealing with in order to figure out how long to energize the coil to get the ball to a specific velocity.

===Industrial Application===

One of the most common industrial applications of Magnetic Dipole is the Nuclear Magnetic Resonance spectroscopy. To learn more about NMR, click [http://www.chemguide.co.uk/analysis/nmr/background.html here]

[[File:Nmrs.gif]]
[http://ftp.uspbpep.com/v29240/usp29nf24s0_c761.html#usp29nf24s0_ic07611| Image Source]

Dipole-like fields are found anywhere there is a coil of wire, that is to say in many electronic applications such as electric motors, transformers, chokes, and many other electronic components. The dipole simplification of magnetic field calculations makes it much easier to calculate for these devices although with the prevalence of computer models more accurate calculations can be performed.

==History==

===Discovery===

[[File: Ampereport.png]] [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/|Image Source]

The discovery of magnetic dipoles is mostly attributed to [https://nationalmaglab.org/magnet-academy/history-of-electricity-magnetism/pioneers/andre-marie-ampere/ | André-Marie Ampère]. Ampere began theorizing about magnetic dipoles when he discovered that current loops produce magnetic fields. Initially, he thought that all magnetic fields result from loops of current. This discovery laid the basis of later work that was able to quantify and explain magnetic fields.

Ampere hails form Lyon, France and his early life was shaped by the French Revolution. His father was a judge, and beheaded by the revolutionaries. Ampere found himself unable to continue his education after the death of his father, but eventually found himself able to return to his studies. Ampere was married to Julie Carron, and had one some with her named Jean-Jacques Ampère. Their son would later grow up to be a famous historian and writer.

===Magnetic Monopoles===

Although electrostatics and magnetics have a lot in common, one difference is that magnetic poles cannot exist singly like charges can. Poles always come in N/S pairs. If you were to cut a bar magnet in half, you will get two new magnets with North and South poles.

A magnetic monopole was observed once, by [https://en.wikipedia.org/wiki/Blas_Cabrera | Blas Cabrera], on February 14, 1982 (also known as the Valentine's Day Monopole). However, this experiment has been recreated several times, and no one, not even Cabrera himself, has been able to find another monopole.

===Magnetic North vs Geographic North===

There is a common misconception about the magnetic field of the earth and the geographical north pole. The geographical north pole is home to the magnetic south pole and vise versa. However, the magnetic and geographical poles are not located in the same place on the planet. This means that while a compass is the first thing that comes to mind when people think about navigation, it is not an effective means of navigation when nearing the poles or in the waters of the far north or south. Below in the [[Further reading]] section, there is an in depth article about the positioning of the magnetic and geographical poles, how they came to be, and the effect they have had on the world.

[[File:Eathmagfield.png]][http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/MagEarth.html|Image Source]

== See also ==

[http://www.physicsbook.gatech.edu/Electric_Dipole Electric Dipole]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Long_Straight_Wire Magnetic Field of a Long Straight Wire]

[http://www.physicsbook.gatech.edu/Magnetic_Field_of_a_Loop Magnetic Field of a Loop]

===Further reading===

[http://science.sciencemag.org/content/303/5663/1494 Terahertz Magnetic Response from Artificial Materials]

[http://www.brighthubengineering.com/marine-engines-machinery/43712-what-is-degaussing-of-ships/ Navy Degaussing Methods]

[http://gisgeography.com/magnetic-north-vs-geographic-true-pole/ True North VS Magnetic North]

==References==

[http://www.chemguide.co.uk/analysis/nmr/background.html| More information on NMR]

[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html| Magnetic Dipole Moment]

[https://phys.org/news/2016-08-mysterious-magnetic-monopole.html| More information on magnetic monopoles]

[http://www.symmetrymagazine.org/breaking/2010/12/15/another-year-of-searching-for-magnetic-monopoles| Attempts at proving the existence of a magnetic monopole]

Lorentz Force

2025-11-30T21:26:47Z

Zeynepuzun: Undo revision 47436 by Zeynepuzun (talk)

[[File:Headerlorentz.png|400px|thumb|right|Lorentz force diagram]]
'''Claimed by Xinyan Jiang, Spring 2025'''
== Introduction ==
The **Lorentz force** is a fundamental concept in electromagnetism, describing the force experienced by a charged particle moving through electric and magnetic fields. It is important in understanding the behavior of particles in various physical systems.

Formally, the Lorentz force is expressed as:

<math> \vec{F} = q\vec{E} + q\vec{v} \times \vec{B} </math>

where:
* <math> q </math> is the electric charge of the particle,
* <math> \vec{E} </math> is the electric field,
* <math> \vec{v} </math> is the velocity of the particle,
* <math> \vec{B} </math> is the magnetic field.

This equation shows how electric fields exert forces in the direction of the field, while magnetic fields exert forces perpendicular to both the particle’s motion and the magnetic field lines. The Lorentz force not only underpins the operation of electric motors, generators, and particle accelerators, but also provides a framework for deeper exploration of classical and modern physics.

== The Main Idea ==

The **Lorentz force** is the total electromagnetic force experienced by a charged particle due to electric and magnetic fields. When a particle with charge <math>q</math> moves with velocity <math>\vec{v}</math> in the presence of an electric field <math>\vec{E}</math> and magnetic field <math>\vec{B}</math>, the net force acting on the particle is given by:

<math> \vec{F}_{\text{Lorentz}} = q\vec{E} + q\vec{v} \times \vec{B} </math>

This equation shows two distinct contributions:
* The '''electric force''': <math>q\vec{E}</math>, which acts in the direction (or opposite direction, depending on the sign of the charge) of the electric field.
* The '''magnetic force''': <math>q\vec{v} \times \vec{B}</math>, which acts perpendicular to both the velocity of the particle and the magnetic field.

The direction of the magnetic force is determined using the **right-hand rule** for positive charges, and reverses for negative charges. The **magnitude** of the magnetic component of the force is given by:

<math> F_{\text{mag}} = qvB\sin\theta </math>

where <math>\theta</math> is the angle between <math>\vec{v}</math> and <math>\vec{B}</math>.

In classical physics, the Lorentz force is the foundation of many electromagnetic phenomena, including the motion of charges in fields, current generation in wires, and the operation of electric and magnetic devices.

examples:

[[File:Two_currents_in _the_same_direction.jpeg|400px|center|thumb|same direction currents]]

==Significance==
The Lorentz force complements Maxwell's equations by describing the force on a charged particle in electromagnetic fields. While Maxwell's equations define how fields are generated, the Lorentz force explains how particles respond to those fields.

This law is central to understanding:

Particle motion in fields

Magnetic field generation from currents

Applications such as electromagnets and transformers

However, it does not fully describe collective particle behavior in materials, which requires more complex models like transport equations.

==A Mathematical Model==

The electromagnetic force ''F'' on a charged particle, the Lorentz force (named after the Dutch physicist [http://www.physicsbook.gatech.edu/Hendrik_Lorentz Hendrik A. Lorentz]) is given by:

<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B}</math>

where '''<math>q\vec{E}</math>''' is the electric force contributed by an external electric field and ''' <math>q\vec{v} ⨯ \vec{B}</math>''' is the magnetic force contributed by an external magnetic field. As many applications involve vectors, it is valuable to recognize the resulting directions of '''<math>q\vec{E}</math>''' and ''' <math>q\vec{v} ⨯ \vec{B}</math>''' in relation to the particle's charge and velocity within the environment of applied electric/magnetic fields. The resulting electric force vector will always be towards or opposite to the applied electric field, depending on the sign of the charge. For example, if an electron is in an electric field along the +x direction, the force will point in the -x direction. The magnetic force is always perpendicular to both the direction of motion and the field. The magnetic force on the particle, however, has a direction perpendicular to both the velocity ''v'' of the particle and the magnetic field ''B'', and has a value proportional to ''q'' and to the magnitude of the vector cross product ''' <math>q\vec{v} ⨯ \vec{B}</math>'''. More specifically, the magnitude of the magnetic force equals ''qvBsinθ'' where ''θ'' is the angle between ''v'' and ''B''.

====Motion of a Charged Particle in a Uniform Magnetic Field====

A noteworthy result of the Lorentz force is the motion of a charged particle within a uniform magnetic field as the angle between ''v'' and ''B'' varies. If a scenario presents a charged particle with a velocity vector ''v'' perpendicular to the applied magnetic field ''B'' (i.e. θ = 90°), the particle will follow a circular trajectory. The radius of this trajectory can easily be calculated:

<math> \vec{F}_{Mag} = \vec{F}_{Centripetal}</math>

<math> qvB = mv^2/R </math>

<math> R = mv/qB </math>

Suppose that for the following example, a charge is shot into a region filled with a uniform magnetic field coming out of the page:

[[File:Meeds1.png]]

At every instant, the external magnetic field ''B'' points out of the page and thus invokes a magnetic force on the particle perpendicular to the particle's velocity - the force needed to create circular motion. The radius ''R'' can be calculated from the equation above. In addition, it's worth noting that the particle's charge will determine where the particle veers. If the particle were positively charged, the magnetic force would cause the particle to veer downward (due to right hand rule), and vice-versa if negatively charged.

In the case that θ is less than 90°, the particle's trajectory will orbit a helix path with a central axis parallel to the field lines.

If θ is zero, that is, the magnetic field is in the same direction as the particle's velocity, the particle will experience no magnetic force and continue to move normally along the field lines.

==A Computational Model==

[https://trinket.io/embed/glowscript/43e56e9c64 Here is a visualization] on VPython of a negatively charged particle moving through a constant electric and magnetic field:

[[File:Lorentzdiagram.png]]

Initially, a negatively charged particle is traveling with initial velocity in the -z direction. There is a constant electric field ''E'' in the -x direction and a constant magnetic field ''B'' in the +y direction. The magnetic force on the negatively charged particle is equal to <math> q\vec{v} ⨯ \vec{B}</math>, or the charge of the particle times the cross product of the particle’s velocity and the magnetic field it travels through. The electric force on the particle is equal to <math> q\vec{E} </math>, or the charge of the particle times the electric field that the particle travels through. Since the magnetic force on the particle is related to the particle’s velocity ''v'', the magnetic force changes as the the particle’s velocity changes. Conversely, the electric force on the particle is constant. Since the Magnetic force is variable, the Lorentz Force on the particle, or the net force due to magnetic and electric forces on the particle (<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B}</math>) is also variable, and the particle's velocity changes.

==Example Problems==

===Simple===

The electric force on a certain particle is <100,-600,300> N and the magnetic force is <-600,400,0> N. Find the Lorentz force.

'''Solution: Lorentz force = <-500,-200,300> N'''
[[File:Soln2.PNG]]

===Intermediate===

The magnetic force on a proton is 100 N at an angle 30 degrees down from the +x axis. The electric force on the proton is 100 N at an angle 30 degrees up from the +z axis. What is the magnitude of the Lorentz Force on the proton?

'''Solution: Lorentz force = 122.5 N'''

[[File:Soln1.PNG]]

===Difficult===
An electron is traveling with a constant velocity of <0.75c, 0, 0>. You measure the magnetic field to be <0.4, 0.3, 0.5>T everywhere. What is the electric field?

'''Solution: E = <0, 1.13e8, -6.75e7> N/C '''

Since the electron is traveling at constant velocity, the net force must be zero. Thus, the magnetic field must equal the electric field, or <math> q\vec{v} ⨯ \vec{B}= q\vec{E} </math>. The charge on both sides cancels out to give <math> \vec{v} ⨯ \vec{B}= \vec{E} </math>. Calculating
<math> q\vec{v} ⨯ \vec{B}</math> = <0, -1.13e8, 6.75e7>, so the electric field must point in the opposite direction.

==Connectedness==

1. Speakers use the Lorentz force of an electromagnet to move a cone that creates sound waves in the air. When current flows through the wires in the electromagnetic in different quantities, the speakers move in unique ways to produce the different sounds that we recognize. Amplifiers for electric guitars and basses work in the same way.

2. In today's evolving world, one area of particular interest is sustainable and renewable energy. Wind turbines and hydropower plants work by harnessing the kinetic energy of wind or water and using it to induce an electrical current. The turbines rotate and move a permanent magnet that induces a current in an electromagnet placed inside of the magnet, which is shaped like a hollow cylinder. The induced current is then carried via wires to external sources to provide energy.

3. Several industries manufacture products that induce current using the Lorentz Force. For example, electric guitars and basses work by magnetizing the strings and relying on the Lorentz force to create a current in pickups that is then transmitted to an amplifier. Pickups are small electromagnet coils surrounding a magnet that are placed beneath the strings. The strings become magnetized because of the magnet inside the pickup. When they are played and vibrate, they induce current in the electromagnet. The Lorentz force causes the strings to exert forces that move mobile charges and induce the current. The current is then increased through a potentiometer and sent to an amplifier through a cable. In addition, charged particle accelerators like cyclotrons make use of the circular orbit particles experience when ''v'' and ''B'' are perpendicular to each other. For each revolution, a carefully timed electric field offers additional kinetic energy to cause the particles to move in increasingly-larger orbits until the desired energy level is met. These particles are known to be extracted and used in a number of ways, from basic studies of the properties of matter to the medical treatment of cancer.

==Applications==

Cyclotrons and particle accelerators exploit the Lorentz force to guide and accelerate particles. When \vec{v} is perpendicular to \vec{B}, particles orbit in circles and gain energy through electric fields.

Used in:

Particle physics (e.g., CERN experiments)

Radiation therapy

Mass spectrometry

More:
Specific applications of the Lorentz force have enabled some of the most significant scientific experimentation to date.

It is known that when the angle between ''v'' and ''B'' is 90 degrees, there is no magnetic force on the particle which continues to move unaffected along the field lines. Cyclotrons and other charged particle accelerators make use of this fact and that particles move in circular orbit when ''v'' and ''B'' are perpendicular with respect to each other (therefore their cross product is equal to 0). With each revolution, these particles gain additional kinetic energy from the electric field which therefore increases orbit. Once reaching the desired energy level, they can be extracted and used in various ways from studies on subatomic particles to even medical treatments for diseases like cancer.

Scientists at the European Organization for Nuclear Research(CERN), the largest organization for particle research, take advantage of the Lorentz force in their collider experiments[4].

==History==
[[[[File:Hendrik Antoon Lorentz.jpg|thumb|right|Hendrik Antoon Lorentz]]

The Lorentz Force is named after Hendrik Lorentz, who derived the formula in the late 19th century following a previous derivation by [[Oliver Heaviside]] in 1889. However, scientists had tried to find formulas for one electromagnetic force for over a hundred years before.Some scientists such as [[Henry Cavendish]] argued that the magnetic poles of an object could create an electric force on a particle that obeys an inverse-square law. However, the experimental proof was not enough to definitively publish. In 1784, [[Charles de Coulomb]], using a torsion balance, was able to definitively show through experiment that this was true. After [[Hans Christian Ørsted]] discovered that a magnetic needle is acted on by a voltaic current, [[Andre Marie Ampere]] derived a new formula for the angular dependence of the force between two current elements. However, the force was still given in terms of the properties of the objects involved and the distances between, not in terms of electric and magnetic fields or forces.

[[Michael Faraday]] introduced modern ideas of magnetic and electric fields, including their interactions and relations with each other, later to be given full mathematical description by [[William Thomson (Lord Kelvin)]] and [[James Maxwell]]. From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents, however, it was not initially evident how his equations related to the forces on moving charged objects. [[J.J. Thomson]] was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as <math>\vec{F} = q\vec{E} + \frac{q}{2}\vec{v} ⨯ \vec{B}</math>. Finally, Heaviside and later Lorentz were able to combine the information into the currently accepted Lorentz Force equation.

== See also ==
The [[Hall Effect]] is a special case in which the magnetic and electric forces on a particle or object cancel out, meaning that there is zero net force. Solving these problems involves setting the two forces equal to each other and using given information to find values for <math>\vec{B}</math>, <math>\vec{v}</math>, or <math>\vec{E}</math>.

[https://www.youtube.com/watch?v=8QWB8IfNoIs This video] demonstrates a few everyday applications and examples of the Lorentz Force.

===Further reading===

[[Hall Effect]]
*http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html
*http://www.ittc.ku.edu/~jstiles/220/handouts/section%203_6%20The%20Lorentz%20Force%20Law%20package.pdf

===External links===

[1]*http://jnaudin.free.fr/lifters/lorentz/
[2]*https://nationalmaglab.org/education/magnet-academy/watchplay/interactive/lorentz-force
[3]*http://web.mit.edu/sahughes/www/8.022/lec10.pdf
[4]*https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.lorentz_force

==References==

*Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3 vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2.: volume 2.
*Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X.
*Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics. Belmont, [CA.]: Thomson Brooks/Cole. ISBN 0-534-40846-X.
*Hughs, Scott. “Magnetic Force; Magnetic Fields; Ampere's Law.” MIT.edu. Magnetic Force; Magnetic Fields; Ampere's Law, 29 Nov. 2017, Boston, MIT, Magnetic Force; Magnetic Fields; Ampere's Law.

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

Lorentz Force

2025-11-30T20:55:03Z

Zeynepuzun:

[[File:Headerlorentz.png|400px|thumb|right|Lorentz force diagram]]
'''Claimed by Zeynep Uzun, Fall 2025'''
== Introduction ==
The **Lorentz force** is a fundamental concept in electromagnetism, describing the force experienced by a charged particle moving through electric and magnetic fields. It is important in understanding the behavior of particles in various physical systems.

Formally, the Lorentz force is expressed as:

<math> \vec{F} = q\vec{E} + q\vec{v} \times \vec{B} </math>

where:
* <math> q </math> is the electric charge of the particle,
* <math> \vec{E} </math> is the electric field,
* <math> \vec{v} </math> is the velocity of the particle,
* <math> \vec{B} </math> is the magnetic field.

This equation shows how electric fields exert forces in the direction of the field, while magnetic fields exert forces perpendicular to both the particle’s motion and the magnetic field lines. The Lorentz force not only underpins the operation of electric motors, generators, and particle accelerators, but also provides a framework for deeper exploration of classical and modern physics.

== The Main Idea ==

The **Lorentz force** is the total electromagnetic force experienced by a charged particle due to electric and magnetic fields. When a particle with charge <math>q</math> moves with velocity <math>\vec{v}</math> in the presence of an electric field <math>\vec{E}</math> and magnetic field <math>\vec{B}</math>, the net force acting on the particle is given by:

<math> \vec{F}_{\text{Lorentz}} = q\vec{E} + q\vec{v} \times \vec{B} </math>

This equation shows two distinct contributions:
* The '''electric force''': <math>q\vec{E}</math>, which acts in the direction (or opposite direction, depending on the sign of the charge) of the electric field.
* The '''magnetic force''': <math>q\vec{v} \times \vec{B}</math>, which acts perpendicular to both the velocity of the particle and the magnetic field.

The direction of the magnetic force is determined using the **right-hand rule** for positive charges, and reverses for negative charges. The **magnitude** of the magnetic component of the force is given by:

<math> F_{\text{mag}} = qvB\sin\theta </math>

where <math>\theta</math> is the angle between <math>\vec{v}</math> and <math>\vec{B}</math>.

In classical physics, the Lorentz force is the foundation of many electromagnetic phenomena, including the motion of charges in fields, current generation in wires, and the operation of electric and magnetic devices.

examples:

[[File:Two_currents_in _the_same_direction.jpeg|400px|center|thumb|same direction currents]]

==Significance==
The Lorentz force complements Maxwell's equations by describing the force on a charged particle in electromagnetic fields. While Maxwell's equations define how fields are generated, the Lorentz force explains how particles respond to those fields.

This law is central to understanding:

Particle motion in fields

Magnetic field generation from currents

Applications such as electromagnets and transformers

However, it does not fully describe collective particle behavior in materials, which requires more complex models like transport equations.

==A Mathematical Model==

The electromagnetic force ''F'' on a charged particle, the Lorentz force (named after the Dutch physicist [http://www.physicsbook.gatech.edu/Hendrik_Lorentz Hendrik A. Lorentz]) is given by:

<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B}</math>

where '''<math>q\vec{E}</math>''' is the electric force contributed by an external electric field and ''' <math>q\vec{v} ⨯ \vec{B}</math>''' is the magnetic force contributed by an external magnetic field. As many applications involve vectors, it is valuable to recognize the resulting directions of '''<math>q\vec{E}</math>''' and ''' <math>q\vec{v} ⨯ \vec{B}</math>''' in relation to the particle's charge and velocity within the environment of applied electric/magnetic fields. The resulting electric force vector will always be towards or opposite to the applied electric field, depending on the sign of the charge. For example, if an electron is in an electric field along the +x direction, the force will point in the -x direction. The magnetic force is always perpendicular to both the direction of motion and the field. The magnetic force on the particle, however, has a direction perpendicular to both the velocity ''v'' of the particle and the magnetic field ''B'', and has a value proportional to ''q'' and to the magnitude of the vector cross product ''' <math>q\vec{v} ⨯ \vec{B}</math>'''. More specifically, the magnitude of the magnetic force equals ''qvBsinθ'' where ''θ'' is the angle between ''v'' and ''B''.

====Motion of a Charged Particle in a Uniform Magnetic Field====

A noteworthy result of the Lorentz force is the motion of a charged particle within a uniform magnetic field as the angle between ''v'' and ''B'' varies. If a scenario presents a charged particle with a velocity vector ''v'' perpendicular to the applied magnetic field ''B'' (i.e. θ = 90°), the particle will follow a circular trajectory. The radius of this trajectory can easily be calculated:

<math> \vec{F}_{Mag} = \vec{F}_{Centripetal}</math>

<math> qvB = mv^2/R </math>

<math> R = mv/qB </math>

Suppose that for the following example, a charge is shot into a region filled with a uniform magnetic field coming out of the page:

[[File:Meeds1.png]]

At every instant, the external magnetic field ''B'' points out of the page and thus invokes a magnetic force on the particle perpendicular to the particle's velocity - the force needed to create circular motion. The radius ''R'' can be calculated from the equation above. In addition, it's worth noting that the particle's charge will determine where the particle veers. If the particle were positively charged, the magnetic force would cause the particle to veer downward (due to right hand rule), and vice-versa if negatively charged.

In the case that θ is less than 90°, the particle's trajectory will orbit a helix path with a central axis parallel to the field lines.

If θ is zero, that is, the magnetic field is in the same direction as the particle's velocity, the particle will experience no magnetic force and continue to move normally along the field lines.

==A Computational Model==

[https://trinket.io/embed/glowscript/43e56e9c64 Here is a visualization] on VPython of a negatively charged particle moving through a constant electric and magnetic field:

[[File:Lorentzdiagram.png]]

Initially, a negatively charged particle is traveling with initial velocity in the -z direction. There is a constant electric field ''E'' in the -x direction and a constant magnetic field ''B'' in the +y direction. The magnetic force on the negatively charged particle is equal to <math> q\vec{v} ⨯ \vec{B}</math>, or the charge of the particle times the cross product of the particle’s velocity and the magnetic field it travels through. The electric force on the particle is equal to <math> q\vec{E} </math>, or the charge of the particle times the electric field that the particle travels through. Since the magnetic force on the particle is related to the particle’s velocity ''v'', the magnetic force changes as the the particle’s velocity changes. Conversely, the electric force on the particle is constant. Since the Magnetic force is variable, the Lorentz Force on the particle, or the net force due to magnetic and electric forces on the particle (<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B}</math>) is also variable, and the particle's velocity changes.

==Example Problems==

===Simple===

The electric force on a certain particle is <100,-600,300> N and the magnetic force is <-600,400,0> N. Find the Lorentz force.

'''Solution: Lorentz force = <-500,-200,300> N'''
[[File:Soln2.PNG]]

===Intermediate===

The magnetic force on a proton is 100 N at an angle 30 degrees down from the +x axis. The electric force on the proton is 100 N at an angle 30 degrees up from the +z axis. What is the magnitude of the Lorentz Force on the proton?

'''Solution: Lorentz force = 122.5 N'''

[[File:Soln1.PNG]]

===Difficult===
An electron is traveling with a constant velocity of <0.75c, 0, 0>. You measure the magnetic field to be <0.4, 0.3, 0.5>T everywhere. What is the electric field?

'''Solution: E = <0, 1.13e8, -6.75e7> N/C '''

Since the electron is traveling at constant velocity, the net force must be zero. Thus, the magnetic field must equal the electric field, or <math> q\vec{v} ⨯ \vec{B}= q\vec{E} </math>. The charge on both sides cancels out to give <math> \vec{v} ⨯ \vec{B}= \vec{E} </math>. Calculating
<math> q\vec{v} ⨯ \vec{B}</math> = <0, -1.13e8, 6.75e7>, so the electric field must point in the opposite direction.

==Connectedness==

1. Speakers use the Lorentz force of an electromagnet to move a cone that creates sound waves in the air. When current flows through the wires in the electromagnetic in different quantities, the speakers move in unique ways to produce the different sounds that we recognize. Amplifiers for electric guitars and basses work in the same way.

2. In today's evolving world, one area of particular interest is sustainable and renewable energy. Wind turbines and hydropower plants work by harnessing the kinetic energy of wind or water and using it to induce an electrical current. The turbines rotate and move a permanent magnet that induces a current in an electromagnet placed inside of the magnet, which is shaped like a hollow cylinder. The induced current is then carried via wires to external sources to provide energy.

3. Several industries manufacture products that induce current using the Lorentz Force. For example, electric guitars and basses work by magnetizing the strings and relying on the Lorentz force to create a current in pickups that is then transmitted to an amplifier. Pickups are small electromagnet coils surrounding a magnet that are placed beneath the strings. The strings become magnetized because of the magnet inside the pickup. When they are played and vibrate, they induce current in the electromagnet. The Lorentz force causes the strings to exert forces that move mobile charges and induce the current. The current is then increased through a potentiometer and sent to an amplifier through a cable. In addition, charged particle accelerators like cyclotrons make use of the circular orbit particles experience when ''v'' and ''B'' are perpendicular to each other. For each revolution, a carefully timed electric field offers additional kinetic energy to cause the particles to move in increasingly-larger orbits until the desired energy level is met. These particles are known to be extracted and used in a number of ways, from basic studies of the properties of matter to the medical treatment of cancer.

==Applications==

Cyclotrons and particle accelerators exploit the Lorentz force to guide and accelerate particles. When \vec{v} is perpendicular to \vec{B}, particles orbit in circles and gain energy through electric fields.

Used in:

Particle physics (e.g., CERN experiments)

Radiation therapy

Mass spectrometry

More:
Specific applications of the Lorentz force have enabled some of the most significant scientific experimentation to date.

It is known that when the angle between ''v'' and ''B'' is 90 degrees, there is no magnetic force on the particle which continues to move unaffected along the field lines. Cyclotrons and other charged particle accelerators make use of this fact and that particles move in circular orbit when ''v'' and ''B'' are perpendicular with respect to each other (therefore their cross product is equal to 0). With each revolution, these particles gain additional kinetic energy from the electric field which therefore increases orbit. Once reaching the desired energy level, they can be extracted and used in various ways from studies on subatomic particles to even medical treatments for diseases like cancer.

Scientists at the European Organization for Nuclear Research(CERN), the largest organization for particle research, take advantage of the Lorentz force in their collider experiments[4].

==History==
[[[[File:Hendrik Antoon Lorentz.jpg|thumb|right|Hendrik Antoon Lorentz]]

The Lorentz Force is named after Hendrik Lorentz, who derived the formula in the late 19th century following a previous derivation by [[Oliver Heaviside]] in 1889. However, scientists had tried to find formulas for one electromagnetic force for over a hundred years before.Some scientists such as [[Henry Cavendish]] argued that the magnetic poles of an object could create an electric force on a particle that obeys an inverse-square law. However, the experimental proof was not enough to definitively publish. In 1784, [[Charles de Coulomb]], using a torsion balance, was able to definitively show through experiment that this was true. After [[Hans Christian Ørsted]] discovered that a magnetic needle is acted on by a voltaic current, [[Andre Marie Ampere]] derived a new formula for the angular dependence of the force between two current elements. However, the force was still given in terms of the properties of the objects involved and the distances between, not in terms of electric and magnetic fields or forces.

[[Michael Faraday]] introduced modern ideas of magnetic and electric fields, including their interactions and relations with each other, later to be given full mathematical description by [[William Thomson (Lord Kelvin)]] and [[James Maxwell]]. From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents, however, it was not initially evident how his equations related to the forces on moving charged objects. [[J.J. Thomson]] was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as <math>\vec{F} = q\vec{E} + \frac{q}{2}\vec{v} ⨯ \vec{B}</math>. Finally, Heaviside and later Lorentz were able to combine the information into the currently accepted Lorentz Force equation.

== See also ==
The [[Hall Effect]] is a special case in which the magnetic and electric forces on a particle or object cancel out, meaning that there is zero net force. Solving these problems involves setting the two forces equal to each other and using given information to find values for <math>\vec{B}</math>, <math>\vec{v}</math>, or <math>\vec{E}</math>.

[https://www.youtube.com/watch?v=8QWB8IfNoIs This video] demonstrates a few everyday applications and examples of the Lorentz Force.

===Further reading===

[[Hall Effect]]
*http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.html
*http://www.ittc.ku.edu/~jstiles/220/handouts/section%203_6%20The%20Lorentz%20Force%20Law%20package.pdf

===External links===

[1]*http://jnaudin.free.fr/lifters/lorentz/
[2]*https://nationalmaglab.org/education/magnet-academy/watchplay/interactive/lorentz-force
[3]*http://web.mit.edu/sahughes/www/8.022/lec10.pdf
[4]*https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.lorentz_force

==References==

*Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics (3 vol.). Pearson / Addison-Wesley. ISBN 0-8053-9047-2.: volume 2.
*Jackson, John David (1999). Classical electrodynamics (3rd ed.). New York, [NY.]: Wiley. ISBN 0-471-30932-X.
*Serway, Raymond A.; Jewett, John W., Jr. (2004). Physics for scientists and engineers, with modern physics. Belmont, [CA.]: Thomson Brooks/Cole. ISBN 0-534-40846-X.
*Hughs, Scott. “Magnetic Force; Magnetic Fields; Ampere's Law.” MIT.edu. Magnetic Force; Magnetic Fields; Ampere's Law, 29 Nov. 2017, Boston, MIT, Magnetic Force; Magnetic Fields; Ampere's Law.

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