Interface conditions for electromagnetic fields

Remarks

Conditions 1 and 2 come from applying the divergence theorem to Gauss's laws on a thin pillbox that straddles $\Sigma$. Conditions 3 and 4 come from applying Stokes' theorem to Faraday's law and the Ampère–Maxwell law on a thin rectangle that crosses $\Sigma$. As the thickness tends to zero, the time derivative terms drop out because $\partial_t B$ and $\partial_t D$ are bounded, while the surface sources remain. So the normal part of $B$ and the tangential part of $E$ are always continuous. The jumps of the displacement field $D$ and of $H$ are given by the surface sources.

Together with the constitutive relations of the two media, these conditions fix how fields are refracted at a boundary. Example: consider the boundary of two dielectrics with permittivities $\varepsilon_1$ and $\varepsilon_2$ and no free surface charge. Let a static field line meet $\Sigma$ at angle $\theta_1$ to the normal and leave at angle $\theta_2$. Condition 3 gives $\abs{E_1} \sin \theta_1 = \abs{E_2} \sin \theta_2$, and condition 1 gives $\varepsilon_1 \abs{E_1} \cos \theta_1 = \varepsilon_2 \abs{E_2} \cos \theta_2$. Dividing, $\tan \theta_1 / \tan \theta_2 = \varepsilon_1 / \varepsilon_2$.

At a perfect conductor the time-varying fields inside vanish, and the conditions become $n \times E = 0$ and $n \cdot B = 0$ on the outside, the usual boundary conditions for metallic waveguides and cavities.

Going deeper

For a student

Interface conditions are the jump conditions of the first-order system of Maxwell's equations, just as the Rankine–Hugoniot conditions are the jump conditions of a conservation law. They turn problems with several media into boundary value problems on each side, coupled through $\Sigma$. From them one derives the Fresnel equations for reflection and refraction of plane waves, and the boundary conditions used in the analysis of waveguides, resonant cavities and scattering problems.