On the Foundational Equations of the Classical Theory of Electrodynamics

A close examination of the Maxwell-Lorentz theory of electrodynamics reveals that polarization and magnetization of material media need not be treated as local averages over small volumes - volumes that nevertheless contain a large number of electric and/or magnetic dipoles. Indeed, Maxwell's macroscopic equations are exact and self-consistent mathematical relations between electromagnetic fields and their sources, which consist of free charge, free current, polarization, and magnetization. When necessary, the discrete nature of the constituents of matter and the granularity of material media can be handled with the aid of special functions, such as Dirac's delta-function. The energy of the electromagnetic field and the exchange of this energy with material media are treated with a single postulate that establishes the Poynting vector S = E × H as the rate of flow of electromagnetic energy under all circumstances. Similarly, the linear and angular momentum densities of the fields are simple functions of the Poynting vector that can be unambiguously evaluated at all points in space and time, irrespective of the type of material media, if any, that might reside at various locations. Finally, we examine the Einstein-Laub force- and torque-density equations, and point out the consistency of these equations with the preceding postulates, with the conservation laws, and with the special theory of relativity. The set of postulates thus obtained constitutes a foundation for the classical theory of electrodynamics.