# Question about the definition of magnetostatics

From my understanding, magnetostatics is defined to be the regime in which the magnetic field is constant in time. However, Griffiths defines magnetostatics to be the regime in which currents are "steady," meaning that the currents have been going on forever and charges aren't allowed to pile up anywhere. This part of Griffiths's definition about charges not being allowed to pile up anywhere seems to be placing a constraint on the meaning of magnetostatics that my initial definition doesn't entail. How do you reconcile these (at least ostensibly) not-equivalent definitions? Does charge being locally conserved have something to do with it?

• Consider what the electric field of charges piling up will do. Is the electric field of such a piling up constant? Oct 31 '18 at 4:52
• @Triatticus Intuitively, no, the electric field wouldn't be constant, but perhaps the piling up in some places compensates for the piling up in other places in such a way that the field remains constant. If someone could prove otherwise that would be nice.
– user113773
Oct 31 '18 at 11:44
• It turns out that the electric field is not constant, but the magnetic field does remain exactly constant (even when factoring in induction effects). See my answer. Nov 8 '18 at 3:14

In this case, Gauss's law for electricity gives that the charge density $$\rho$$ must be time-independent as well. But then the continuity equation $${\bf \nabla} \cdot {\bf J} = -\partial \rho/\partial t$$ implies that the current field must be divergence-free. (Conceptually, the subtlety is that if you had steady currents that led to charge piling up, then the charge buildup would create a time-dependent electric field, which would in turn induce another contribution to the magnetic field.)