A nonlinear prerelativistic approach to mathematical representation of vacuum electromagnetism (original) (raw)
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This paper presents an alternative to the Maxwell vacuum equations pre-relativistic approach to description of electromagnetic field objects. Our view is based on the understanding that the corresponding differential equations should be dynamical in nature and the physical relations represented by them should represent local stress-energy-momentum balance relations. Such a view does not go along with the classical assumption for local recognizability of the electric and magnetic constituents E and B as time-stable and space propagationg subsystems of the field objects. The corresponding reconsideration brought us to the assumption, that the two couples (E, B) and (−B, E) are much more adequate in this respect: free electromagnetic field objects exist in a permanent propagation with the fundamental velocity c, so each of its recognizable subsystems should be able to carry momentum, and neither E nor B are able to do this separately, while each of the couples (E, B; −B, E) is able to do this, but only in presence of the other. Therefore, the necessary internal local dynamics, admissible changes, time stability and recognizability during space propagation should be viewed in terms of (E, B) and (−B, E) and their mutually compatible changes.
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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.