Electric interface condition for sliding and viscous contacts

Abstract

The first principles of electromagnetism impose that the tangential electric field must be continuous at the interface between two media. The definition of the electric field depends on the frame of reference, leading to an ambiguity in the mathematical expression of the continuity condition when the two sides of the interface do not share the same rest frame. We briefly review the arguments supporting each choice of interface condition and illustrate how the most theoretically consistent choice leads to a paradox in induction experiments. We then present a model of sliding contact between two solids and between a fluid and a solid and show how this paradox can be lifted by taking into account the shear induced by the differential motion in a thin intermediate viscous layer at the interface, thereby also lifting the ambiguity in the electric interface condition. We present some guidelines regarding the appropriate interface condition to employ in magnetohydrodynamics applications, in particular for numerical simulations in which sliding contact is used as an approximation of the viscous interface between a conducting solid and a fluid of very low viscosity such as in planetary interior simulations.