What can Maxwell's Equations tell us about permanent magnets/ how are permanent magnets and electromagnets related?

Question

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.

Answer 1

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.