On the proper formulation of Maxwellian electrodynamics for continuum mechanics

Abstract

Despite the importance of electromagnetomechanical physics to processes ranging from piezoelectricity to the dynamics of electron beams, confusion abounds in the continuum mechanics literature as to how Maxwell’s equations of electrodynamics should be formulated in the material frame of continuum mechanics. Current formulations in the literature conflict as to the manner in which the authors define fields, derive constitutive relations, and interpret contradictory formulations. The difficulties persist even when the phenomena described are electrostatic. This paper will demonstrate that the perplexity arises from two sources: a misunderstanding of the limitations of material frame descriptions, and the failure to appreciate the centrality of relativity theory to the formulation of electrodynamic equations in the vicinity of mechanical motion. Two new formulations of Maxwell’s equations are provided that avoid the paradoxes of earlier formulations and thus describe the physics clearly and without self-contradiction.