Abstract
A fundamental result of classical electromagnetism is that Maxwell’s equations imply that electric charge is locally conserved. Here we show the converse: Local charge conservation implies the local existence of fields satisfying Maxwell’s equations. This holds true for any conserved quantity satisfying a continuity equation. It is obtained by means of a strong form of the Poincaré lemma presented here that states: Divergence-free multivector fields locally possess curl-free antiderivatives on flat manifolds. The above converse is an application of this lemma in the case of divergence-free vector fields in spacetime. We also provide conditions under which the result generalizes to curved manifolds.
Highlights
Maxwell’s equations of electromagnetism have taken various forms
The fact that the continuity equation follows from Maxwell’s equations is a fundamental result of electrodynamics, and it is crucial to the theory, because it means that electric charge is locally conserved
We have presented two main results: a strong form of the Poincare lemma (Theorem 4.2) and its application to conserved currents (Theorem 5.1)
Summary
Maxwell’s equations of electromagnetism have taken various forms. They were first proposed by James Clerk Maxwell in 1865 as a set of twenty equations [13]. The very last of these, Maxwell called the equation of continuity, in analogy to the equation of mass continuity in hydrodynamics. In his original treatise, this was written as de df dg dh + + + = 0. Griffith’s Introduction to Electrodynamics (2012), you will find the familiar four equation expression for Maxwell’s equations: This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECCUNICAMP, Campinas, Brazil, edited by Sebastia Xambo-Descamps and Carlile Lavor
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