Magnetic vector potential and electric scalar potential in three-dimensional eddy current problem

In calculating eddy currents in a conductor by means of the vector potential for which the Coulomb Gauge is used, the scalar potential appears when the electric conductivity varies in the conductor, while it is not necessary for the case of the constant electric conductivity. As the field equation is a Helmholtz-type equation under the Coulomb Gauge condition, we must determine the boundary conditions for the scalar potential and each of the components of the vector potential. How to determine the boundary conditions is presented, together with an illustrative example of a two-dimensional eddy current problem with the variable electric conductivity.