Suppose we fix an inertial frame of reference and we have a set of electric charges which do not move, i.e. $$\pmb J(\pmb r,t)=0$$ $$\frac{\partial\rho}{\partial t}=0$$

Maxwell equations do not rule out the possibility of there being a non-zero magnetic field, so my questions essentialy are

  1. I have been taught that to generate magnetic field you need a current...is this at least partially incorrect, in light of the above observation?
  2. Could you show a physically significant example of a magnetic field arising out of non-moving charges?
  • $\begingroup$ The solution to all four maxwell equations being satisfied simultaneously is known as Jefimenko's Equations. Plugging your specified conditions on them, yields null magnetic field. $\endgroup$ – Physicist137 Jan 3 '18 at 1:49

Yes, non-moving charges do produce magnetic fields, but you won't get a full explanation of why until you study advanced quantum mechanics (specifically, quantum field theory as it relates to quantum electrodynamics). The most common charged particles in the universe are protons and neutrons, and they have spin $1/2$, guaranteeing that they will always have a non-zero magnetic moment, producing a magnetic dipole field, even when stationary. This spin is not motion of the charge in any sense, it's an "internal" rotation of the Fermion's field.

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  • $\begingroup$ Are you implying that it is not possible within classical electromagnetism? $\endgroup$ – Nicol Jan 3 '18 at 0:21
  • $\begingroup$ @Nicol I wouldn't imply it, I'd outright say it. In classical electromagnetism the permanent magnetic moment that leads to things like ferromagnets just has to be take as a given that is a bit mysterious. Once you accept their existence, though, you can work with them. Classically, you'd expect exactly that limitation that only moving charges produce magnetic fields, and hence the slightly misleading name "spin" when there isn't some spinning sphere of charge underlying the phenomenon. $\endgroup$ – Sean E. Lake Jan 3 '18 at 0:27
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    $\begingroup$ Ohanian in "What is spin?", Am. J. Phys. 54 (6), June 1986 writes "on the basis of an old calculation by Belinfante [ Physica 6, 887 ( 1939) ] , it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are "internal"- $\endgroup$ – hyportnex Jan 3 '18 at 13:54
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    $\begingroup$ [... ]they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of· angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave" $\endgroup$ – hyportnex Jan 3 '18 at 13:54

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