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?

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    $\begingroup$ Consider what the electric field of charges piling up will do. Is the electric field of such a piling up constant? $\endgroup$
    – Triatticus
    Oct 31 '18 at 4:52
  • $\begingroup$ @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. $\endgroup$
    – user113773
    Oct 31 '18 at 11:44
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    $\begingroup$ 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. $\endgroup$
    – tparker
    Nov 8 '18 at 3:14

Like all questions about definitions, the answer is not fundamentally "interesting" because, well, it all just boils down to your choice of definitions. But I would define "magnetostatics" to be the regime in which neither the magnetic field nor the electric field depends on time. The motivation for this definition is that the behavior of classical E&M changes very qualitatively (and becomes much more complicated) once the fields become time-dependent, so the regime in which neither field changes over time is a natural one to consider separately.

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.)

If you require the current to only be time-independent but not divergenceless, then it turns out that the resulting magnetic field is also constant in time and given exactly by the Biot-Savart law. But you can have steady charge buildup, and the electric field is given by Coulomb's law with the charges evaluated at the present time, not the retarded time - a situation that may appear to, but doesn't actually, violate causality. This is a rather subtle situation, and whether or not you consider it to be "magnetostatic" is a matter of personal preference - but most people probably wouldn't, because even though the magnetic field is constant in time, it still has a contribution from the time-changing electric field via Ampere's law.

  • $\begingroup$ This was a very useful answer, thank you! Would you know of any resources discussing the subtle situation you mention in your final paragraph? $\endgroup$
    – 1729_SR
    Nov 3 '20 at 13:31
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    $\begingroup$ @1729_SR Yes: aapt.scitation.org/doi/abs/10.1119/1.16589 $\endgroup$
    – tparker
    Nov 3 '20 at 14:21

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