Classical Physics


Maxwell Field Equation Tutorial: Its Causal Lorentz-Covariant Solution in Integral Form

Authors: Steven Kenneth Kauffmann

This tutorial sets out the detailed steps which produce the causal solution in integral form of the Maxwell field equation set, the causal solution being the one which vanishes in the absence of the source charge-current density, including during the time prior to that source being switched on. After warming up by obtaining the causal solution in integral form for the static source and field case, the Lorentz-covariant antisymmetric electromagnetic field tensor is defined, and the Maxwell field equation set is converted to its Lorentz-covariant form, whose formal causal solution is obtained in nine lines, but whose Green's function initially is ill-defined, a subtle issue whose ultimate causal resolution prompts a substantial effusion of ink. The integral solution which emerges can be exhibited either as manifestly Lorentz covariant or as very closely related to the static-case solution, except for systematic causal time retardation. Although direct solution for the electromagnetic field tensor is emphasized, the algebraically less involved four-vector intermediary potential approach is outlined in parallel.

Comments: 5 Pages.

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Submission history

[v1] 2019-03-06 14:04:44

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