Thermodynamics of Electrical Networks and the Onsager-Casimir Reciprocal Relations

Abstract

Electrical networks furnish an excellent example for the application of thermodynamics of irreversible processes. In particular, they provide one of the simplest cases where, besides Onsager coefficients, one has Casimir coefficients in the phenomenological equations. Two thermodynamic formulations of the network equations are given, which closely resemble the Lagrangian and the Hamiltonian formalism, respectively, of classical mechanics. In the first formulation, only Onsager coefficients occur, but the thermodynamic forces are of a peculiar type in that they are Lagrangian derivatives. Incidentally, it is shown that Casimir reciprocal relations can generally be replaced by Onsager reciprocal relations if the independent variables in the linear phenomenological relations are chosen in a proper way. As a generalization of the network equations, Maxwell's equations in continuous matter with dielectric, magnetic, and Joulean heat losses are considered. Matter is assumed to be isothermal, but not necessarily uniform nor isotropic. Under the influence of impressed electric fields, current distributions are produced. The connection of these fields is expressed by a generalized admittance function. A well-known reciprocity theorem for electromagnetic fields is seen to hold even if all types of losses, as mentioned before, are present. This is due to the Onsager-Casimir reciprocal relations for the dielectric tensor, the permeability tensor, and the resistivity tensor. From the reciprocity theorem, a symmetry relation can be derived for the generalized admittance function. A generalized version is given in the presence of a static magnetic field.