Notes on Magnetostatic Interactions in $\mathbb R^3$

Abstract

A compact formulation of magnetostatics is presented. Without using constitutive relations, including the aether relations, an expression is proposed for the potential energy of a region carrying a distribution of magnetic dipoles under a vector potential field, where the dipole distribution is specified by a current density potential field. Assuming that during a virtual motion of the dipole distribution the virtual work expended by the field is equal to minus the time derivative of the potential energy, an expression for the mechanical force functional is derived. In addition to the Lorentz and Kelvin force densities, the force functional contains an asymmetric active stress field, the skew-symmetric part of which is the mechanical couple density, and a pressure term.