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It makes sense that Maxwell's equations tell us that there are no monopoles, but can the equations tell us anything else about the magnetic fields of permanent magnets on their own, i.e. without interactions with a wire/current, or how such fields arise?

I only have a facile understanding of Maxwell's equations and I was wondering if someone who knows more than me can elaborate a bit. Permanent magnets and electromagnets must be intimately related somehow, but it seems a lot of the introductory literature emphasizes their differences.

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  • $\begingroup$ Permanent magnets have to do with the "frozen" alignement of the involved subatomic particles. For electromagnets the induction of magnetic field is overshadows by the discovery, that moving in a coil electrons producing magnetic fields. The deeper understanding that for accelerated in a circle (and this is what electrons are involved in a coil) electrons their intrinsic spins and the related magnetic dipole moments are aligned and the common magnetic field is the sum of all this aligned electrons is not reflected well until now. So your question is very reasonable. $\endgroup$ Sep 27 '16 at 16:19
  • $\begingroup$ See my answer to the question What is the nature of magnetic fields? $\endgroup$ Sep 27 '16 at 16:20
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    $\begingroup$ @HolgerFiedler if you have the time, you could expand your comment into an answer, as imo, it's an important question that does not seem to be duplicated anywhere. I think it is different to your link above. It's title could really be written as, How do we duplicate natural magnets by using electromagnets.? $\endgroup$
    – user108787
    Sep 27 '16 at 17:20
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but can the equations tell us anything else about the magnetic fields of permanent magnets on their own, i.e. without interactions with a wire/current, or how such fields arise?

Permanent magnets have a nonzero magnetization $\textbf{M}$ which gives rise to bound volume and surface currents $\textbf{J}_b=\nabla\times\textbf{M}$ and $\textbf{K}_b=\textbf{M}\times\textbf{n}$ which in turn contributes to a vector potential $\textbf{A}(\textbf{r})$. Maxwell's equation (without displacement current) $$\nabla\times \textbf{B}=\mu_0\textbf{J}$$ then, in principle, gives the magnetic field. There is nothing in classical electrodynamics that is beyond Maxwell's equations and the Lorentz force law.

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