Electric displacement, polarization and boundary conditions when there's a current flowing

Question

I'm studying from Reitz's Foundations of Electromagnetic Theory and trying to understand how the results obtained under electrostatic conditions are affected when there is a current flowing through material.

I understand that it isn't true anymore that the electric field inside a conductor is zero if there is a current going through the conductor. And I think (please correct me if I'm mistaken) that because of that, now the electric field doesn't need to be perpendicular to the surface of the conductor. So I guess this may affect the boundary conditions, but how are they affected?

Specifically, how do the electric displacement and the polarization behave when there is a current flowing from one material with some permitivity $\epsilon_1$ and conductivity $g_1$ to another one with permitivity $\epsilon_2$ and conductivity $g_2$? Is it still true that a free charge density is accumulated at the surface between both materials?

I mean, is this boundary condition: $$(\vec{D_1}-\vec{D_2}) \hat n =\sigma$$ still valid?

I'm quite confused, because I don't get what it would mean to have a surface charge when all the charge is actually flowing.

Answer 1

In the case of an interface of dielectrics with conductivity the application of an electric field leads to the build up of a free interface charge which modifies the normal fields so that current continuity holds at the interface, i.e., $$\vec{J_1}\hat n = g_1\vec{E_1}\hat n=g_2\vec{E_2}\hat n = \vec{J_2}\hat n$$ Your boundary condition for the interface charge is correct. See my answers to the similar questions here: Free charge in a dielectric and Surface charge density in conducting plate .

Note: The interface charge builds up in the first moment upon applying an electric field to the interface due to the different conductivities and at first equal fields. Very quickly a steady state situation arises where you have a stationary interface charge and equal current densities entering and leaving the interface.