Relation between magnetic fields and electric currents in plasmas

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/4 pi)del x B. Here I investigate the relation between del x 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 (Vasyliunas, 2001): 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 lambda(e), a given del x B produces the corresponding J, while a given J does not produce any del x B but disappears instead. The reason for this can be understood by noting that del x B not equal(4 pi/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 lambda(e).) On larger scales, the evolution of B (and hence also of del x B) is governed by del x 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.