The way back: from charge conservation to Maxwell equations

Abstract

The main purpose of this article is to disseminate among a wide audience of physicists a known result, which is available since a couple of years to the \emph{cognoscenti} of differential forms on manifolds; namely, that charge conservation implies the inhomogeneous Maxwell equations. This is the reciprocal statement of one which is very well known among physicists: charge conservation, written in the form of a continuity equation, follows as a consequence of Maxwell equations. We discuss the conditions under which charge conservation implies Maxwell equations. The key role played by the constitutive equations is hereby stressed. The discussion is based on Helmholtz theorem, according to which a vector field is determined by its divergence and its curl. Green's functions are also shown to play a fundamental role. We present all results in three-vector, as well as in tensorial notation. We employ only those mathematical tools most physicists are familiar with.