When Maxwell postulated the existence of displacement currents to arrive at a nonstatic

Chapter 7, Problem 17

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When Maxwell postulated the existence of displacement currents to arrive at a nonstatic version of Amp`eres law, he was simply choosing the simplest way to correct equation (22) so that it would be consistent with the continuity equation (23). However, other possibilities are also consistent with the continuity equation. (a) Show that in order to have equation (22) valid in the static case, then, in general, we must have B = 0J + F1 t for some (time-varying) vector field F1 of classC2. (b) By taking the divergence of both sides of the equation in part (a), show that F1 t = 0 0 E t . (c) Use part (b) to argue that, from an entirely mathematical perspective, Amp`eres law can also be generalized as B = 0J + 0 0 E t + F2, where F2 is any divergence-free vector field. Since no one has observed any physical evidence for F2s being nonzero, it is assumed to be zero, as in equation (24).