Question

A conductor with resistivity ρ bounds on a dielectric with permittivity ε. At a certain point A at the conductor's surface the electric displacement equals D, the vector D being directed away from the conductor and forming an angle α with the normal of the surface. Find the surface density of charges on the conductor at the point A and the current density in the conductor in the vicinity of the same point.

Answer 1

The dielectric ends in a conductor. It is given that on one side (the dielectric side) the electric displacement D is as shown. Within the conductor, at any point A, there can be no normal component of electric field. For if there were such a field, a current will flow towards depositing charge there which in turn will set up countering electric field causing the normal component to vanish. Then by Gauss theorem, we easily derive

σ = D_n = D cos α where a is the surface charge density at A. 

The tangential component is determined from the circulation theorem

It must be continuous across the surface of the conductor. Thus, inside the conductor, there is a tangential electric field of a magnitude,

This implies a current, by Ohm’s law, of