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 


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*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?

*Could you show a physically significant example of a magnetic field arising out of non-moving charges?

 A: 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 electrons, 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. Neutrons also have this property because they are built from charged particles. This spin is not motion of the charge in any sense, it's an "internal" rotation of the Fermion's field.
A: Non-moving charges with well-defined position and momentum cannot produce a magnetic field. Classically, current and charge density are the only sources of the electromagnetic field, and while we can calculate a magnetic field associated with a changing electric field, it is clear that an electric field will only change in time when the charge density is changing. The other source of a magnetic field would be current density, obviously ruled out. (This all assumes an isolated system, i.e. we neglect any boundary conditions.)
Note that Maxwell's equations are field equations (Eulerian) so the motion of charge is only represented by the change in charge density per second. Conversely, electrodynamic theories of the electron (i.e. theories which take point charges as their basis) are Lagrangian in nature, they describe the position and velocity of a particle over time. But because this requires knowledge of the point charge's position and momentum exactly, at every instant, such a description is mostly at odds with reality, unless we change our meaning of a "point charge" to "a charge density with a monopole moment and zero multipole moments which is much smaller than the system being considered." In other words, if the charge density is really small and symmetric, we can consider it as a point charge.
Anywho, that digression aside, the only reason Maxwell's equations in differential form do not rule out the possibility of a magnetic field even when $\partial \rho/\partial  t = \mathbf J = 0$, is because they allow for external sources, outside of the system, which would be causing that magnetic field. But that would, in turn, require moving charges.
To understand foundationally the origin of permanent magnetics, ferromagnetism, etc, would require a theory of matter which is outside the scope of classical electrodynamics, and in any case it gives rise to situations where quantum effects cannot be neglected and classical electrodynamics no longer holds. But we can model such instances within the framework of classical physics by treating them as equivalent external current densities or charge distributions.
