Maxwell’s and Wave Equations in Media Containing Electromagnetic Field Sources

Abstract

This paper presents a new system of Maxwell’s equations and corresponding wave equations for conducting media containing electromagnetic field sources. The theoretical approach begins with a consistent generalisation of Maxwell’s differential equations in a slowly moving conducting medium. From this generalised form, both differential and integral forms of Maxwell’s equations are derived for slowly moving and stationary conducting media. The well-known Maxwell’s equations for a perfect dielectric containing sources appear as a special case of the equations for a conducting medium containing sources. Notably, Maxwell’s third differential equation is derived from the first, resulting in a new form of the third equation specific to conducting media. This revised equation differs from the traditional form valid in perfect dielectrics – namely, Gauss’s law for the electric field – which holds in a conducting medium only when the medium is linear, isotropic, homogeneous and source-free. A key contribution of this work is the introduction of the volume density of the source leakage electric current in conducting media as a novel physical quantity, analogous to the volume charge density in perfect dielectrics. The volume density of the leakage electric current injected by the source into the surrounding medium is reduced to the volume density of the displacement electric current in the case of a dielectric, where the conducting component is absent. Finally, wave equations for electromagnetic fields and electromagnetic potentials are derived from the generalised Maxwell’s differential equations.