# Can non-moving charges produce a magnetic field?

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?
• 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. – 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.