# Magnetic moment and current of permanant magnet

Every magnetic dipole has a magnetic moment associated to it. Magnetic moment of electromagnets could be found by ##M = iA##. But for permanant magnets how it is measured? If we use this formula it implies that there must be some internal current flowing in permanant magnet.
Is it true?
What exactly is the difference between electromagnet and permanant magnet with respect to current flowing through them?

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Simon Bridge
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Is it true?
No it isn't.

Every current has a magnetic field associated with it but not every magnetic field has a current.

No it isn't.

Every current has a magnetic field associated with it but not every magnetic field has a current.
Then what causes magnetic field in permanant magnets?

Simon Bridge
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marcusl
Gold Member
It is also true, however, that a permanent magnet can be replaced or modeled by an effective surface current density that produces the same fields outside.

Khashishi
Electrons have spin, which is associated with magnetic field.

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A follow-on to my post #6 above. Detailed calculations show that the magnetic fields computed from the magnetic surface currents are in precise agreement with those calculated by the magnetic pole method, and in addition, the magnetic surface current theory actually explains much of the underlying physics. In addition the OP asks how does the current flow? The magnetic moments from the individual atoms do result in currents, but for a uniform distribution of them, essentially adjacent currents cancel. A simple analogy illustrating this is to consider the squares of a checkerboard to each contain an electron orbiting in the same direction (e.g. clockwise) on the edges of each individual square. The currents from adjacent orbits precisely cancel and the net effect is a current on the outer edge of the whole checkerboard. Notice with this surface current that there is no actual charge transport around the outer edge of the board. This is very much the nature of these magnetic surface currents. The atomic model is very consistent with the magnetic surface current per unit length formula ## K_m=M \times \hat{n} /\mu_o ##. For a permanent magnet of cylindrical geometry, the magnetic surface currents are around the outer cylindrical surface, directly analogous to the currents in a solenoid. Normally, the magnetic surface currents(per unit length) in a permanent magnet are about 100 times stronger than a typical DC current (per unit length) in a solenoid. The magnetic field of a permanent cylindrical magnet has geometrically the same shape (both inside and outside) as that of a bare solenoid but will thereby typically be about 100 times stronger.

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Khashishi
Permanent magnets are made of materials that have unpaired electrons with spins which become aligned at low enough temperature. So the magnetic field is a superposition of many individual spin magnetic moments. Ampere's law states that there should be a current in a permanent magnet, but it cannot be a classical flowing current. So I guess Ampere's law is not correct for quantum objects (unless you introduce a new kind of "quantum current" within each electron).

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Permanent magnets are made of materials that have unpaired electrons with spins which become aligned at low enough temperature. So the magnetic field is a superposition of many individual spin magnetic moments. Ampere's law states that there should be a current in a permanent magnet, but it cannot be a classical flowing current. So I guess Ampere's law is not correct for quantum objects (unless you introduce a new kind of "quantum current" within each electron).
The equation that applies on a macroscopic level is ## \nabla \times M=\mu_o J_m ##. For uniform magnetization ## M ## , there are no magnetization currents (##\nabla \times M=0 ##). At any surface boundary, the curl becomes discontinuous and with Stokes theorem, the magnetic surface current per unit length is ## K_m=M \times \hat{n}/\mu_o ##. Magnetization ## M ## is defined as the number of magnetic moments per unit volume ## n ##, (assuming they point in the same direction) times the magnetic moment ## \mu=I \cdot A ##. ## \ ## ##M=n\mu ##. Biot-Savart and/or Ampere's law can be used to compute the ## B ## field from the magnetic surface currents. Even though there is no actual charge transport in magnetic surface currents, the calculations with Biot-Savart and Ampere's law still apply. Please see also the "link" to the homework question in post #6. I gave the OP who posted the homework problem a detailed response.

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Homework Helper
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Excellent question - did you try looking it up?
... also see "intrinsic magnetic moment".

You can associate a current with electron "orbits" but that tends to give you the wrong picture of how electrons behave. It is certainly different from the idea of having a current in the magnet in the sense of a bulk circulation of charge like in a loop of wire.
This "link" is quite interesting, but the consensus there from a couple of the postings is that noone seems to understand how a simple cylindrical magnet works. The magnetic surface current approach, mentioned in post #5 by @marcusl that I also mentioned in post #6 provides not only a qualitative explanation, but also a precise quantitative explanation. The calculations for the magnetic field ## B ## from the (macroscopic) magnetization vector ## M ## in the material are quite straightforward and can be performed without the quantum details of the origins of the individual magnetic moments. Particularly for a uniform magnetization ## M ## of a cylindrical permanent magnet, the magnetic surface current calculations and the magnetic field ## B ## that arises from them can readily be performed by most first and second year college physics students and it really should be part of the curriculum. Griffith's E&M textbook introduces the magnetic surface currents, but his explanation, using the magnetic vector potential ## A ##, is rather mathematical and the magnetic surface currents that arise from his derivation can easily be overlooked.

Simon Bridge
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Except that, then, magnetism is just a material property that you have to measure rather than something derived from fundamental interactions.
The thing is, however compelling the model, there is no physical macroscopic current in a permanent magnet giving rise to the magnetism. The "surface current" thing is a representation, not a physical reality.

The best you can do is invoke the atomic magnetic dipole arising from the angular momenta of it's electrons.

Bear in mind that "we don't know" is an acceptable answer in science. It is possible to know what is false without knowing what is actually true.
As it happens there are very good models for how magnetism arises from fundamental interactions.
Did you try looking up "intrinsic magnetic moment"?

davenn
davenn
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The thing is, however compelling the model, there is no physical macroscopic current in a permanent magnet giving rise to the magnetism. The "surface current" thing is a representation, not a physical reality
and from my various readings, that is also what I have come to understand

Dave

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Except that, then, magnetism is just a material property that you have to measure rather than something derived from fundamental interactions.
The thing is, however compelling the model, there is no physical macroscopic current in a permanent magnet giving rise to the magnetism. The "surface current" thing is a representation, not a physical reality.

The best you can do is invoke the atomic magnetic dipole arising from the angular momenta of it's electrons.

Bear in mind that "we don't know" is an acceptable answer in science. It is possible to know what is false without knowing what is actually true.
As it happens there are very good models for how magnetism arises from fundamental interactions.
Did you try looking up "intrinsic magnetic moment"?
I am familiar with the connection between the (quantum mechanical) angular momentum and the intrinsic magnetic moment, and also the concept of electron spin. I do think if you would perform a few calculations with the magnetic surface currents you would be surprised at how consistent the calculations are with the "magnetostatic pole model" which is emphasized in J.D. Jackson's Classical Electrodynamics. Alternatively, Griffith's E&M text presents the surface current method and does a lengthy mathematical derivation involving the magnetic vector potential ## A ## for a distribution of magnetic dipoles. The result is a formula that includes the magnetic surface current term along with a term that involves ## \nabla \times M ##. (Basically the result that ## \nabla \times M=\mu_o J_m ## with the surface current formula ## K_m=M \times \hat{n}/\mu_o ## resulting from the application of Stokes theorem.) I have personally performed quite a number of calculations (much more extensive than those shown in the "link" of post #6 above) that show complete consistency between the magnetic field ## B ## computed from the "pole" and "surface current" models . The argument can be made that the surface currents are not real because there is no actual charge transport. When considering transformers and the laminated layers to block the currents from the Faraday EMF in the iron, the puzzle arises, if the magnetization works from surface currents, how can it work if the laminations would block the surface currents? And the answer is that the laminations do not affect the surface currents, because no charge transport is necessary, but the "eddy" currents from the Farday EMF are blocked because they involve electrical charge transport. I do encourage you to not discount the magnetic surface currents and give them a more thorough study. Everyone is entitled to their own concepts, but for me, the magnetic surface current theory added a tremendous amount of clarity to the "pole" model, and calculations that previously contained a lot of handwaving (using the pole method), were now well founded in first principles and generated the exact same results for the ## B ## fields. The magnetic surface currents even provide a simple explanation for the previously unexplained shape of the M vs. H hysteresis curves and the process of reversing the direction of magnetization in a permanent magnet. (The solenoid current per unit length must be equal in amplitude and opposite the magnetic surface current per unit length.) The magnetic surface current theory explains much of how a permanent magnet works as well as other features of magnetism. If anyone is at all interested, I'd be glad to show more complete detail of calculations supporting the magnetic surface current theory, and as a PM if that is more suitable.

Simon Bridge
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I do think if you would perform a few calculations with the magnetic surface currents you would be surprised at how consistent the calculations are with the... etc etc
No I won't be - I have performed more than "a few" calculations that way and it is not at all surprising etc etc etc.
So what?

OP asked what caused the magnetism in permanent magnets - it is not caused by "surface currents".

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No I won't be - I have performed more than "a few" calculations that way and it is not at all surprising etc etc etc.
So what?

OP asked what caused the magnetism in permanent magnets - it is not caused by "surface currents".
I am of the opinion that the magnetism in permanent magnets is most readily explained by magnetic surface currents. By magnetism, I believe the OP is referring to the magnetic fields. The magnetic fields that result from the magnetic surface currents (which comes from the magnetization) maintain the magnetization in a permanent magnet ,keeping the individual magnetic moments aligned, in a self-consistent way. (along with the quantum exchange effect which essentially gives them a strong tendency to cluster in the same direction.) The individual magnetic moments respond to the magnetic field ## B ## with energy ##E=-\mu \cdot B ## and thereby will have lowest energy when they are aligned with the magnetic field. The exchange effect, to first order, has the effect of clustering the magnetic dipoles together in a unit considerably larger than a single electron spin.

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Khashishi
Fitzpatrick claims that the surface currents arise from adding together all the microscopic atomic currents.
http://farside.ph.utexas.edu/teaching/302l/lectures/node77.html
But, I have a problem with his description because he only mentions the orbital effect of the electrons in the sentence "Well, atoms consist of negatively charged electrons in orbit around positively charged nuclei." He doesn't mention the spin magnetic moment of the electrons, which are a big part of the overall magnetic moment and Lande g-factor. I don't think the spin magnetic moment can be viewed as an atomic current without some license of the imagination.

jtbell
Mentor
You need to keep in mind that the notion of "bound currents" (surface and volume) in describing magnetized materials originated in classical electrodynamics, before the rise of quantum mechanics and spin and orbital angular momentum. We still teach it as part of classical electrodynamics because it's useful for calculations in situations where an atomic-level description based on QM would (presumably) not produce significantly different results, and would be much more complicated to calculate with.

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Fitzpatrick claims that the surface currents arise from adding together all the microscopic atomic currents.
http://farside.ph.utexas.edu/teaching/302l/lectures/node77.html
But, I have a problem with his description because he only mentions the orbital effect of the electrons in the sentence "Well, atoms consist of negatively charged electrons in orbit around positively charged nuclei." He doesn't mention the spin magnetic moment of the electrons, which are a big part of the overall magnetic moment and Lande g-factor. I don't think the spin magnetic moment can be viewed as an atomic current without some license of the imagination.
For orbital angular momentum, the "g" factor is 1, where magnetic moment ## \mu=g_L u_B L/(h/(2\pi)) ## . The z-angular momentum contained in the magnetic surface current is consistent with the z-angular momentum totaled up from all the individual magnetic moments. The equation ## \mu=g_S u_B S/(h/(2 \pi)) ## with g_S=2 (almost precisely) is one complicating factor that I can't explain away. Perhaps there is a simple explanation for it, and it's possible that there isn't one. The macroscopic models (pole model and surface current models) assume uniform magnetic fields for cases of uniform magnetization with no variation in the fields at the atomic level. For this reason, they clearly have their limitations, but the magnetic surface current model offers some very good explanations at the macroscopic level. Even though it may fail to explain everything, it is both sufficiently consistent and mathematically accurate that I think it should be emphasized in the undergraduate E&M physics curriculum, perhaps with the disclaimer that it doesn't explain everything.

Dale
Mentor
The "surface current" thing is a representation, not a physical reality
In some sense that is true of everything in scientific theories. The "bound current" is a perfectly reasonable classical representation, and it is frequently used and taught. A quantum model may be more accurate, but often unnecessarily complicated.

Simon Bridge
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Yes - it is a perfectly reasonable representation ... I have not disagreed with that.
To your knowledge, has anyone measured the surface current on a bar magnet that causes it's magnetic field?
(i.e. other than deducing it from the presence of the field?)

Anyway: surface current model for the beginning student:
http://farside.ph.utexas.edu/teaching/302l/lectures/node77.html
... at this level the net surface current gets caused by the atomic surface currents due to orbits of electrons.

The next level is to use the atomic dipoles rather than the currents... though the bridging is usually something like:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/orbmag.html
... starting from a planetary model of the atom with circular orbits.

I suppose it depends on where the student is going... chemistry students get by with a more literal idea of "orbitals" than physics students for eg.
Here's an accessible version in terms of magnetic moment density... still classical.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/magpr.html

jtbell
Mentor
The derivation that leads to the definitions of bound surface and volume currents starts with the magnetic vector potential ##\vec A## of a single ideal pointlike dipole, assumes that an infinitesimal volume element dV of the material has a dipole moment ##\vec M dV## (where ##\vec M## varies with position, in general), and integrates over the volume of the material to find the total ##\vec A##.

It turns out that the total ##\vec A## has two terms. If you make the substitution ##\vec J_b = \vec \nabla \times \vec M## in one term, you get an expression that is identical with the ##\vec A## from a volume current density ##\vec J_b##. If you make the substitution ##\vec K_b = \vec M \times \hat n## in the other term, you get an expression that is identical with the ##\vec A## from a surface current density ##\vec K_b##. You can see the details in section 6.2.1 of Griffiths's textbook (3rd edition). He then says:

Griffiths said:
What this means is that the potential (and hence also the field) of a magnetized object is the same as would be produced by a volume current ##\vec J_b = \vec \nabla \times \vec M## throughout the material, plus a surface current ##\vec K_b = \vec M \times \hat n## on the boundary.
(I added the boldface for emphasis.)

The description in terms of little current loops that Fitzpatrick uses (as does Griffiths, in the section following his derivation) is a heuristic device for making the derivation plausible in a pictorial way. The actual mathematical derivation makes no assumption about the nature of the microscopic dipole moments inside the material.

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Griffith's derivation is an extremely good one, but because the result is presented in terms of the magnetic vector potential ## A ## instead of the magnetic field ## B ##, I do think a number of the E&M students are likely to overlook the significance of this derivation. From the vector potential ## A ##, the magnetic field ## B ## can be computed. The results are that for a distribution of magnetization ## M ##, the currents ## J_b=\nabla \times M/\mu_o ## and ## K_b=M \times \hat{n}/\mu_o ## become sources for the magnetic field ## B ## using the Biot-Savart equation.

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Homework Helper
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The derivation that leads to the definitions of bound surface and volume currents starts with the magnetic vector potential ##\vec A## of a single ideal pointlike dipole, assumes that an infinitesimal volume element dV of the material has a dipole moment ##\vec M dV## (where ##\vec M## varies with position, in general), and integrates over the volume of the material to find the total ##\vec A##.

It turns out that the total ##\vec A## has two terms. If you make the substitution ##\vec J_b = \vec \nabla \times \vec M## in one term, you get an expression that is identical with the ##\vec A## from a volume current density ##\vec J_b##. If you make the substitution ##\vec K_b = \vec M \times \hat n## in the other term, you get an expression that is identical with the ##\vec A## from a surface current density ##\vec K_b##. You can see the details in section 6.2.1 of Griffiths's textbook (3rd edition). He then says:

(I added the boldface for emphasis.)

The description in terms of little current loops that Fitzpatrick uses (as does Griffiths, in the section following his derivation) is a heuristic device for making the derivation plausible in a pictorial way. The actual mathematical derivation makes no assumption about the nature of the microscopic dipole moments inside the material.
Thank you @jtbell . I do think this section of Griffith's textbook should be given additional emphasis in the physics curriculum. And as I mentioned in post #23, I think this important result can be so easily overlooked.

marcusl