A Variable Metric Electrodynamics. The Coulomb and Biot–Savart Laws in Anisotropic Media

Abstract

The use of differential forms allows the formulation of the principal equations of electrodynamics in a metric independent way. At this stage a wealth of directed quantities is necessary. The metric is needed only for finding the solutions. After a metric is established, all directed quantities can be replaced by vectors and pseudovectors. Various metrics can be introduced, depending on the medium. We show that a special metric, connected with the electric permittivity tensor, is useful in finding the counterpart of the Coulomb law in an anisotropic dielectric. Another metric, related to the magnetic permeability tensor, helps to find the generalizations of the Biot-Savart law in an anisotropic magnetic medium. The use of special metric allows us to reduce all electrostatic and magnetostatic problems in anisotropic media to those in the isotropic one.