Refraction of the electric field lines, at the interface of separation between two conductive media

Question

Suppose we have 2 media with electrical parameters ${\varepsilon _1},\,{\sigma _1}$, respectively ${\varepsilon _2},\,{\sigma _2}$, separated by the plane surface $\Sigma $; electrical charge surface density on $\Sigma $ is ${\rho _s} = 0$.We denote by $\overrightarrow {{E_1}} ,\,\overrightarrow {{E_2}} $ the electric field vectors in the two environments, and by $\overrightarrow {{J_1}} ,\,\overrightarrow {{J_2}} $ the corresponding current density, with ${J_1} = {\sigma _1} \cdot {E_1}$ and ${J_2} = {\sigma _2} \cdot {E_2}$. Let ${\alpha _1},\,{\alpha _2}$ be the angles between the normal n at the surface $\Sigma $ and the vectors $\overrightarrow {{E_1}} ,\,\overrightarrow {{E_2}} $.

By assuming the boundary conditions ${E_{t1}} = {E_{t2}} , {D_{n1}} = {D_{n2}}$, we obtain the the refractive condition of the electric field lines ${\varepsilon _1} \cdot tg\left( {{\alpha _2}} \right) = {\varepsilon _2} \cdot tg\left( {{\alpha _1}} \right)$.

By assuming the boundary conditions ${E_{t1}} = {E_{t2}} , {J_{n1}} = {J_{n2}}$, we obtain the the refractive condition of the electric field ${\sigma _1} \cdot tg\left( {{\alpha _2}} \right) = {\sigma _2} \cdot tg\left( {{\alpha _1}} \right)$; the boundary condition ${J_{n1}} = {J_{n2}}$ is obtained from the electric current continuity equation.

Because the values of ${\varepsilon _1},\,{\sigma _1}$ and ${\varepsilon _2},\,{\sigma _2}$ are arbitrary material parameters, this yield different values of the angle of refraction of the electric field for the two boundary conditions set. What is the explanation of this paradox?

Answer 1

The explanation of your paradox is that the boundary condition Dn1 = Dn2 does not hold in the case of current flowing across the boundary. There is a (free) sheet charge 𝜂 at the interface so that the electric displacement becomes discontinuous Dn2 - Dn2 = 𝜂. The correct normal electric field boundary condition is σ1⋅En1 = σ2⋅En2 as deduced from the normal current continuity Jn1 = Jn2. This causes a discontinuity of the normal dielectric displacement und thus the interface sheet charge 𝜂. The build-up of 𝜂 can be thought of to be caused by different normal interface current densities before the situation settles in steady state. Thus the above second refractive condition of the electric field lines has to be used at interfaces of conducting media.