The relationship between Electromagnetism and Special Relativity is something that always amazed me but I haven't fully understood yet.
Misconceptions and contradictions seem to be widespreaded, and they don't make the whole picture easy to understand if your background isn't strong enough. Sometimes you hear "They're two sides of the same thing, as for space and time, or mass and energy", or that "Magnetism is just electricity seen from another frame of reference", or "Magnetism is just a quanum effect as well as electricity".
As far as I'm concerned, They could be all true, or......not.
I've watched the well-known series about it made by Veritasium and Minutephysics.
Although I found it very illustrative and well-exposed, I really didn't get the connections and made me want to know more( probably its intent), because topiscs were presented in a intuitive level without caring too much about how or why things behave that way.
The series start with an introductory "Magnetism and Electricity are jut different sides of the same thing", then shows "Magnetism is just electricity from another frame of reference", and concludes "Magnetism is a quantum effect" (In the permanent magnet video).
Especially while presenting Permanent Magnets, the video seems to make different assumptions and make contradictory statements.
It starts by intoducing the spin magnetic moment of a particle addressing it as "a technical mumbo-jumbo to remember us that particle with electric charge have also a magnetic moment", but the wikipedia page on this topic report the opposite "A particle may have a spin magnetic moment without having an electric charge. For example, the neutron is electrically neutral...".
Well it's not quite the opposite, but it prevents us to say that the magnetic moment of a particle is related to charge, hence electricity.
So, I really can't manage to make light on this topic, maybe too adavanced for me.
But curiosity is eating me up and I'd like to know,
"Is the fact that permanent magnets are what they're a proof that magnetism isn't always related to and determined by the electrical properties of a system?",
"Is magnetism really a relativistic effect, or does it have this caratheristic only in a larger scale?",
or "Is magnetism a property of a particle just like its charge, making it a quantum property?"

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    $\begingroup$ There is a discussion by Daniel W. Schroeder titled Magnetism, Radiation, and relativity It's a quantitative discussion, specifically written to be as accessible as possible. Schroeder mentions that the approach he uses goes back to the 60's, published in the Electricity and Magnetism textbook by Edward M. Purcell $\endgroup$
    – Cleonis
    Nov 21 '18 at 20:06
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    $\begingroup$ @Cleonis: Schroeder's notes look nice, but they don't have anything about permanent magnets, which is what this question is about. $\endgroup$
    – user4552
    Nov 21 '18 at 23:28
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    $\begingroup$ One of the common reactions I see from people who have been impressed by an initial read-through of Purcell or a similar treatment is that they think it means more than it really does. E.g., sometimes they get the impression that it's supposed to be possible to reduce any example involving magnetism to a purely electrical example, simply by changing frames of reference. All we really find out from this sort of thing is something about the form of Maxwell's equations. $\endgroup$
    – user4552
    Nov 21 '18 at 23:38
  • $\begingroup$ @BenCrowell thank you for your comment. I was always intrigued by the Purcell treatment, $\endgroup$
    – Cleonis
    Nov 26 '18 at 1:57

The idea that magnetism is electricity viewed from a different frame helps, but is a bit incomplete

What you should really have in mind is that electricity and magnetism are one single phenomenon: electromagnetism

Consider a boost of $v$ along the $x$ axis of a reference frame in which we have electric fields $\mathbf E$ and magnetic fields $\mathbf B$. The fields in the boosted frame become (you can find a proof in most undergrad E&M books, a nice one being Feynman's Lectures)

$$E_x'=E_x$$ $$B_x'=B_x$$ $$E_y'=\gamma\left(E_y-vB_z\right)$$ $$B_y'=\gamma\left(B_y+\frac{v}{c^2}E_z\right)$$ $$E_z'=\gamma\left(E_z+vB_y\right)$$ $$B_z'=\gamma\left(B_z-\frac{v}{c^2}E_y\right)$$

Here $\gamma=\left(1-\frac{v^2}{c^2}\right)^{-\frac{1}{2}}$ is the usual Lorentz factor. It's not as simple as electric fields becoming magnetic. It's more like both are mixing together.

This transformation law is valid for any electromagnetic field, including that of a permanent magnet. If you shoot a permanent magnet really quickly it will generate an electric field (remember Faraday's Law).

This should answer your first two questions. The third one, however is also almost a true statement. Spin magnetic moment is something that arises in Quantum Field Theory. It is to do with the many different ways we have to construct "wavefunctions" for relativistic particles.

When we construct them, we usually need to impose a transformation law under Lorentz transformations, and this gives rise to the quantum number we call spin (since these transformations are matrices and can only have an integer number of dimensions).

By themselves, the particles only end up conserving the sum of total spin and angular momentum, which is why we say spin is a form of angular momentum. However, when we couple these "wavefunctions" to the electromagnetic field, the particles end up reacting to it in the same way as a magnet with magnetic moment proportional to this spin quantum number.

So I wouldn't say that spin is inherently quantum mechanical, but it is inherent to the field theory formalism in which we build Quantum Field Theory.

  • $\begingroup$ QFT enters into play in a fundamental way when the relevant energies are comparable with the mass of particles. However, even for low energy electrons (energy << 0.5 MeV) spin effects are present and well described by Pauli equation which is neither relativistic nor part of QFT. $\endgroup$
    – GiorgioP
    Nov 22 '18 at 0:13
  • $\begingroup$ @GiorgioP true, but then you are postulating spin through its algebra instead of letting it surface from the Lie algebra of Lorentz transformations $\endgroup$ Nov 22 '18 at 1:23

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