Maxwell's equations allow the magnetic field B to be calculated if the electric current density J is assumed to be completely known as a function of space and time. The charged particles that constitute the current, however, are subject to Newton's laws as well, and J can be changed by forces acting on charged particles. Particularly in plasmas, where the concentration of charged particles is high, the effect of the electromagnetic field calculated from a given J on J itself cannot be ignored. Whereas in ordinary laboratory physics one is accustomed to take J as primary and B as derived from J, it is often asserted that in plasmas B should be viewed as primary and J as derived from B simply as (c/4π)∇×B. Here I investigate the relation between ∇×B and J in the same terms and by the same method as previously applied to the MHD relation between the electric field and the plasma bulk flow vmv2001: assume that one but not the other is present initially, and calculate what happens. The result is that, for configurations with spatial scales much larger than the electron inertial length λ_e, a given ∇×B produces the corresponding J, while a given J does not produce any ∇×B but disappears instead. The reason for this can be understood by noting that ∇×B≠4π/c)J implies a time-varying electric field (displacement current) which acts to change both terms (in order to bring them toward equality); the changes in the two terms, however, proceed on different time scales, light travel time for B and electron plasma period for J, and clearly the term changing much more slowly is the one that survives. (By definition, the two time scales are equal at λ_e.) On larger scales, the evolution of B (and hence also of ∇×B) is governed by ∇×E, with E determined by plasma dynamics via the generalized Ohm's law; as illustrative simple examples, I discuss the formation of magnetic drift currents in the magnetosphere and of Pedersen and Hall currents in the ionosphere.

Keywords. Ionosphere (Electric fields and currents) – Magnetospheric physics (Magnetosphere-ionosphere interactions) – Space plasma physics (Kinetic and MHD theory)