  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 consistant 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 where sliding contact is used as an approximation to the
viscous interface between a conducting solid and a fluid of very low viscosity
such as in planetary interior simulations.
