If I understand correctly, the electrostatic approximation assumes that all charges are stationary (i.e the charge density is constant in time, and current density is zero). The magnetostatic approximation assumes the current density is constant in time. Are these the correct definitions?
I have also seen electrostatics being characterized by an electric field which is constant in time. Is this supposed to be some obvious consequence of the above definition?
Looking at Maxwell's equations, the divergence of $E$ is fixed, but the curl of $E$ might vary in time if $B$ varies in time. Even in empty space (an electrostatic situation by defualt) we have the wave solution to Maxwell's equations, in which both $E$ and $B$ vary in time.
So I'm guessing most authors are just sloppy when it comes to nomenclature, and when they say electrostatics, they really mean quasi-electrostatics, i.e electrostatics in the quasistatic approximation where the deplacement current $j_D$ is neglected from Ampere's law.
In quasi-electrostatics, we definitely can't have the wave solution in empty space anymore. But is it still possible that the $B$ field varies in time? The magnetic field equations reduce to $curl(B)=0$ and $div(B)=0$. How do we know it's not possible to find some time-varying $B$ field which always has zero curl and zero divergence?