# What causes the magnetic force in a uniform magnetic field?

• I
• Hak
Hak
As is well known, when electric current flows in a straight wire, a charge in motion in the vicinity of the wire, parallel to it, is affected by a force called the magnetic force, which is attributable to the relativistic contraction effects that the wire undergoes in the frame of reference of the moving charge. I was wondering: in the case of a uniform magnetic field generated by, say, two magnets, what is the relativistic effect (if any, but I think it is because of the similarity of the two phenomena) that causes the charge to feel this force? What "contracts" and what does not? In the case of the wire, we can indeed consider the contraction of the negative charges in motion within them (to simplify the phenomenon), which leads to a different linear charge density and thus to the apparent "magnetic force," which is nothing more than an electric force; in the magnet, what plays the role of these charges in motion?

The doubt really came to me while trying to give a relativistic justification for magnetic phenomena, and I got stuck right at that point. I am afraid, however, that since we are dealing with the usual realm of the infinitely small, not only relativity but perhaps quantum mechanics is involved. The same analogy with the coil confuses me: in the coil you have a magnetic field (I think) precisely because you have a contraction of the distance between the electric charges inside the wire, but when you have, for example, a single elementary charge (an electron) what is contracting?

Any possible and available Insight is appreciated. Thank you very much.

Hak said:
what is the relativistic effect (if any, but I think it is because of the similarity of the two phenomena) that causes the charge to feel this force?
In relativity a permanent magnet is described by the magnetization polarization tensor $$\mathcal{M}^{\mu \nu} = \begin{pmatrix} 0 & P_x c & P_y c & P_z c \\ - P_x c & 0 & -M_z & M_y \\ - P_y c & M_z & 0 & -M_x \\ - P_z c & -M_y & M_x & 0 \end{pmatrix}$$

So when this is boosted, what is a pure magnetization in one frame becomes both a magnetization and electrical polarization in the other frame. This polarization produces an electric field that exerts the force on the charge.

vanhees71 and PeterDonis
Dale said:
In relativity a permanent magnet is described by the magnetization polarization tensor $$\mathcal{M}^{\mu \nu} = \begin{pmatrix} 0 & P_x c & P_y c & P_z c \\ - P_x c & 0 & -M_z & M_y \\ - P_y c & M_z & 0 & -M_x \\ - P_z c & -M_y & M_x & 0 \end{pmatrix}$$

So when this is boosted, what is a pure magnetization in one frame becomes both a magnetization and electrical polarization in the other frame. This polarization produces an electric field that exerts the force on the charge.
Forgive me if I know next to nothing about the subject, I am still a student just starting university. All I knew before this message is that a magnet is generally composed of small elements of matter that have non-zero intrinsic magnetic dipole moments. A simple model for one of these is an atom consisting of a nucleus with a positive charge and electrons revolving around it with a negative charge. This configuration is similar to a current-carrying coil, and it is known that a current-carrying coil generates a magnetic field.
I knew that electric and magnetic fields were basically the same thing in special relativity. In my opinion, those who do not specialise in this field often study the covariant formulation of electromagnetism in vacuum, whereas to interpret magnetised materials, one would need to know how these things work in matter.
Thank you for your competent intervention.

vanhees71
Sagittarius A-Star said:
The magnetic force in a certain frame is not only apparent. The magnetic part of electromagnetism is implied by relativity and it's transformation of E- and B-fields.Yes, see:
https://en.wikipedia.org/wiki/Electron_magnetic_moment

Thank you so much. Could you please tell me which of the posts in the thread to which you referred me (https://www.physicsforums.com/threa...c-field-using-relativity.1056642/post-6959388) are useful or essential to fully understand the question on the topic I have submitted? Thank you.

Hak said:
Thank you so much. Could you please tell me which of the posts in the thread to which you referred me (https://www.physicsforums.com/threa...c-field-using-relativity.1056642/post-6959388) are useful or essential to fully understand the question on the topic I have submitted? Thank you.
That thread is about the specific scenario "electric current flows in a straight wire". But you ask here in the OP for permanent magnets.

My posting there, to which I linked (#10), shows more generally, that the magnetic force is a relativistic effect.

I think that the simplest way to say it is that there is no such thing as the electric field, and there is no such thing as the magnetic field. There is only the electromagnetic field.

What we call the electric and magnetic fields are just aspects of the electromagnetic field, and using different frames implies different ways of splitting the electromagnetic field into electric and magnetic components. That's why different frames "see" different electric and magnetic fields, and that's why saying things like the apparent "magnetic force," which is nothing more than an electric force is wrong. It's always an electromagnetic force. Sometimes it's possible to find a frame where the electromagnetic field has zero magnetic component, but that doesn't mean it's "nothing more than" an electric field. Sometimes you can find a frame where the electromagnetic field has zero electric component, but again that doesn't mean it's "nothing more than" a magnetic field.

Purcell's explanation of the origin of the magnetic field around a wire is great. Similar to learning the resolution of the Twin Paradox, it shows you how a seeming paradox resolves itself with a little care to keep track of every relevant factor. But don't mistake it for a general explanation for magnetism in all circumstances.

martinbn, topsquark, vanhees71 and 1 other person
Ibix said:
I think that the simplest way to say it is that there is no such thing as the electric field, and there is no such thing as the magnetic field. There is only the electromagnetic field.

What we call the electric and magnetic fields are just aspects of the electromagnetic field, and using different frames implies different ways of splitting the electromagnetic field into electric and magnetic components. That's why different frames "see" different electric and magnetic fields, and that's why saying things like the apparent "magnetic force," which is nothing more than an electric force is wrong. It's always an electromagnetic force. Sometimes it's possible to find a frame where the electromagnetic field has zero magnetic component, but that doesn't mean it's "nothing more than" an electric field. Sometimes you can find a frame where the electromagnetic field has zero electric component, but again that doesn't mean it's "nothing more than" a magnetic field.

Purcell's explanation of the origin of the magnetic field around a wire is great. Similar to learning the resolution of the Twin Paradox, it shows you how a seeming paradox resolves itself with a little care to keep track of every relevant factor. But don't mistake it for a general explanation for magnetism in all circumstances.
What is Purcell's explanation? Where can I find it? For example, I am aware of the volumes 'Electricity and Magnetism' by Purcell and Morin, but I do not know where one can find this explanation. If it can be found there, could you tell me the pages under consideration? In general, I would like to know more about this explanation. Thank you.

Hak said:
What is Purcell's explanation? Where can I find it? For example, I am aware of the volumes 'Electricity and Magnetism' by Purcell and Morin, but I do not know where one can find this explanation. If it can be found there, could you tell me the pages under consideration? In general, I would like to know more about this explanation. Thank you.
In the 3rd edition of 'Electricity and Magnetism' by Purcell and Morin: chapter "5.9 Interaction between a moving charge and other charges", pages 259 to 268.

Simplified calculation for a special case:
https://physics.weber.edu/schroeder/mrr/MRRtalk.html

Correction to such a calculation:
http://kirkmcd.princeton.edu/examples/wire.pdf

Ibix and vanhees71
Purcell is pretty confusing. The best book using the relativity-first approach to classical electrodynamics is Landau and Lifshitz vol. 2, which is however at an advanced level.

Ibix and Sagittarius A-Star
vanhees71 said:
The 2nd link is correct. Here's my version of the same thing
https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf
The 1st link calculates only the special case, that the velocity of the test particle is equal to the (small) average velocity of the electrons in the wire. This calculation is probably wrong, because in the electron's frame the wire is neutral.

In Purcell's book the general case for arbitrary velocity of the test particle ##0\le v<c## is calculated. There it can be argued, that for large enough ##v##, it can be practically ignored, that the wire is neutral in the electron's frame and not in the lab frame. In the 2nd link (McDonald) it is estimated, that in the lab frame there are approximately only 10 more electrons than protons in each mm³ copper wire (for an electron velocity of 1cm/s).

McDonald also has a disclaimer:
4Some authors appear to argue that if a system of battery and wire is electrically neutral before the battery is connected (unlikely to be true in practice), with the initial charge densities ρ+ and ρ− being“exactly” equal, it is impossible that this condition changes, even by one part in 1021, after the battery is connected. Some of this debate is reviewed in [40].
Link to the referenced paper [40} by Ron Folman (open access):
https://iopscience.iop.org/article/10.1088/1742-6596/437/1/012013

In this paper, also a superconductor as wire and a possible experiment are discussed.

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Sagittarius A-Star said:
I read part of this examination on Wikipedia: thank you for directing me to it. Do you, by any chance, know of any interesting articles more specific to the problem I have submitted? Thank you very much.

Hak said:
I read part of this examination on Wikipedia: thank you for directing me to it. Do you, by any chance, know of any interesting articles more specific to the problem I have submitted? Thank you very much.
According to the following paper from 1955, also the magnetic moment of the electron seem to be a relativistic effect. Unfortunately, I do not know much about the Dirac equation.
paper said:
The relativistic Dirac theory of the electron assumed a particle that was endowed with the properties of mass and charge. The spin and magnetic moment postulated by Uhlenbeck and Goudsmit were then found to be a consequence of the relativistic invariance of the Dirac equation.
Source:

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Sagittarius A-Star said:
According to the following paper from 1955, also the magnetic momentum of the electron seem to be a relativistic effect. Unfortunately, I do not know much about the Dirac equation.

Source:
Of course you can describe the electron with spin also in non-relativstic QT. It leads to the Pauli equation. You also get the correct gyrofactor 2, if you do the "minimal-substitution heuristics" (i.e., making ##\partial_{\mu} \rightarrow \partial_{\mu} + \mathrm{i} q A_{\mu}## in the Schrödinger equation to get a gauge-invariant coupling between em. field and charged matter).

Sagittarius A-Star said:
The 1st link calculates only the special case, that the velocity of the test particle is equal to the (small) average velocity of the electrons in the wire. This calculation is probably wrong, because in the electron's frame the wire is neutral.

In Purcell's book the general case for arbitrary velocity of the test particle ##0\le v<c## is calculated. There it can be argued, that for large enough ##v##, it can be practically ignored, that the wire is neutral in the electron's frame and not in the lab frame. In the 2nd link (McDonald) it is estimated, that in the lab frame there are approximately only 10 more electrons than protons in each mm³ copper wire (for an electron velocity of 1cm/s).

McDonald also has a disclaimer:

Link to the referenced paper [40} by Ron Folman (open access):
https://iopscience.iop.org/article/10.1088/1742-6596/437/1/012013

In this paper, also a superconductor as wire and a possible experiment are discussed.
Of course, the argument by McDonald (and in my manuscript as well as in the there quoted AJP paper) hinges on the constitutive equation, i.e., Ohm's Law using the usual classical model of conduction electrons a la Drude. It's a fully relativistic model and as such consistent. In this model the wire gets negatively charged due to the "self-consistent" Hall effect with the magnetic field created by the current of the conduction electrons itself, which is the dynamics behind Ohm's Law within the Drude model.

There's of course no contradiction to charge conservation. The charge has to be delivered by the battery.

vanhees71 said:
Of course, the argument by McDonald (and in my manuscript as well as in the there quoted AJP paper) hinges on the constitutive equation, i.e., Ohm's Law using the usual classical model of conduction electrons a la Drude. It's a fully relativistic model and as such consistent. In this model the wire gets negatively charged due to the "self-consistent" Hall effect with the magnetic field created by the current of the conduction electrons itself, which is the dynamics behind Ohm's Law within the Drude model.

There's of course no contradiction to charge conservation. The charge has to be delivered by the battery.

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