Macroscopic Field Equations for Metals in Equilibrium

Abstract

The usual derivation of Maxwell's equations for magnetic materials rests on the assumption that the sources of magnetic field within the material can be split up into a magnetization density M and a current density j. In metals the same electrons (the conduction electrons) contribute both to M and to j, and one is forced to consider the question of what one means by M and what one means by j. In this paper we answer the question for systems in equilibrium, using a thermodynamic approach. The separation of sources of magnetic field into M and j is to a large extent arbitrary, but can be done in such a way that M is uniquely related to the local magnetic field and j is zero for a normal metal in equilibrium, while in the mixed state of a superconductor it satisfies the force-balance equation $\frac{(\mathbf{j}\times \mathbf{B})}{c}+\mathbf{P}=0\mathrm{}$, P being the pinning force. The stress tensor for a magnetic system is derived from first principles (not assuming the field equations), and used to obtain the force-balance equation by an alternative method. Finally, two-dimensional systems such as superconducting thin films and surface sheaths are examined by similar methods.