Classical power and energy relations for macroscopic dipolar continua derived from the microscopic Maxwell equations

Abstract

Positive semi-definite expressions for the time-domain macroscopic energy density in passive, spatially nondispersive, dipolar continua are derived from the underlying microscopic Maxwell equations satisfied by classical models of discrete bound dipolar molecules or inclusions of the material or metamaterial continua. The microscopic derivation reveals two distinct positive semi-definite macroscopic energy expressions, one that applies to diamagnetic continua and another that applies to paramagnetic continua. The diamagnetic dipoles are "unconditionally passive" in that their Amperian magnetic dipole moments are zero in the absence of applied fields. The analysis of paramagnetic continua, whose magnetization is caused by the alignment of randomly oriented "permanent" Amperian magnetic dipole moments that dominate any induced diamagnetic magnetization, is greatly simplified by first proving that the microscopic power equations for rotating "permanent" Amperian magnetic dipoles (which are shown to not satisfy unconditional passivity) reduce effectively to the same power equations obeyed by rotating unconditionally passive magnetic-charge magnetic dipoles. The difference between the macroscopic paramagnetic and diamagnetic energy expressions is equal to a "hidden energy" that parallels the hidden momentum often attributed to Amperian magnetic dipoles. The microscopic derivation reveals that this hidden energy is drawn from the reservoir of inductive energy in the initial paramagnetic microscopic Amperian magnetic dipole moments. The macroscopic, positive semi-definite, time-domain energy expressions are applied to lossless bianisotropic media to determine the inequalities obeyed by the frequency-domain bianisotropic constitutive parameters. Subtleties associated with the causality as well as the group and energy-transport velocities for diamagnetic media are discussed.