A Symmetric Formulation of Maxwell's Equations

Abstract

The electromagnetic field formed by the pair (E, B) obeying Maxwell's equations (MEs) is reformulated as another pair (P, N) obeying a symmetric set of four equations tautologically equivalent to MEs. The symmetric set is formed by a pair of induction and a pair of source equations, each pair with exactly the same structure. In contrast to (E, B), charge and current densities contribute equally to both P and N. The equation of continuity is not an independent condition, but it is automatically fulfilled by any four-tuple (P, N, J, ρ) solving the symmetric MEs. The symmetric equations in terms of potentials may be explicitly solved for a variety of constraint conditions, thus leading to different classes of solutions. Each class represents a family of problems defined by the constraints. One such family is the conventional class of solutions of MEs. Some unexpected results regarding the conventional solutions of MEs in terms of potentials are: (a) it is a particular case, (b) it may contain magnetic scalar potentials, and (c) there is no Coulomb gauge, thus removing the magnetic transversality constraint. It is not known whether the other classes of solutions correspond to physical problems.