Electrodynamics of moving media

Abstract

A. RESEARCH OBJECTIVESFinding the force of electromagnetic origin in a continuous medium is an old problem that has not been solved except in rather special cases. (An exception is the work of Meixner and his associates. -4) It involves the unification of two physical theoriescontinuum mechanics and electromagnetism.Since nonrelativistic mechanics is not, in general, compatible with electromagnetism, relativistic mechanics must be used.The relativistic theory of continuum mechanics (without electromagnetic fields) is fairly well understood, at least insofar as its basic equations are concerned. Electromagnetism in stationary bodies is also fairly well understood.To unite the two theories, the effect of each upon the other must be calculated. The study of the effect of mechanical motion and deformation upon electromagnetism is known as "electromagnetism of moving media," and there are several theories that 5 are formally different but lead to the same physical predictions.The inverse problem, finding the effects of the electromagnetic fields upon the mechanical equations, has never been solved adequately, and it is this problem that we wish to discuss now.The fundamental equations of relativistic continuum mechanics are: (1) Newton's Law, which, in its simplest form, is expressed aswhere n is the number of particles per unit volume, m the relativistic mass per particle, V the velocity, and fmech the force density, now of entirely mechanical origin; and (2) a specification of the way in which fmech is related to n, the strain, the strain rate or other kinematic variables. That is, the fundamental equations are Newton's Law, and the mechanical (and thermodynamic) constitutive relations of the material.The fundamental equations of electromagnetism of stationary bodies are Maxwell's equations, and relations among the field quantities, that is, electromagnetic constitutive relations of the material.It is instructive to consider how mechanical motion affects electromagnetism because the inverse problem, broadly speaking, is similar. Fundamentally, mechanical motion can affect electromagnetism in two ways. First, additional terms may enter, as source terms, into Maxwell's equations. Second, the constitutive relations may be altered in form.There is essentially only one theory of electromagnetism in free space, but, if materials are present, there are several, including (among others) the Minkowski QPR No. 70

Energy - Momentum Tensors for Dispersive Electromagnetic Waves

“…The latter author, while emphasizing the difficulty of obtaining a general microscopic derivation, points out that the problem of obtaining a macroscopic solution in arbitrarily moving media has been solved by Penfield and Haus (1967) using their method of virtual power. As Robinson points out, the result is in fact a generalization of Helmholtz's (1882) solution for the force density acting on a medium subject to static fields, a result which predates the controversy itself by almost 30 years!…”

“…This point has been made clearly by Penfield and Haus (1967). The Minkowski form is wrong only if one demands that the energy-momentum tensor for the background subsystem be unaffected in form by the introduction of interacting fields.…”

“…It is essential to vary the background coordinate field in Hamilton's principle (taking into account such constraints as mass conservation) in order to obtain the correct ponderomotive force; and so a relativistic variational technique has been developed which includes the constraints explicitly, unlike Penfield and Haus (1967) who use Lagrange multipliers.…”

“…We also know some of the EulerLagrange equations a priori, such as Maxwell's equations, and we find that we are unambiguously led to a definite form for the total Lagrangian density. Rules for forming Lagrangian densities are further discussed by Penfield and Haus (1967), who defend Hamilton's principle with the remark that the systematic bookkeeping and standardized set of rules for applying the variational principle allow one to derive equations of motion in a way that is likely to be free of errors.…”

Quantum Limits on Measurement and Control of a Mechanical Oscillator

“…This analogy has been previously used to model moving boundary effects as an effective dielectric perturbation [183]. Finally we note that the conceptual ambiguities associated with identifying the force applied by an electromagnetic field on a moving body [184,185] appear to be less severe in the approach outlined here, wherein the emphasis is on a well-defined observable -cavity frequency -and not on the details of the microscopic light-matter interaction.…”