Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 36B / b / Solution

Let $\hat{n}$ be the unit normal pointing from the minus medium to the plus medium. Integrating the Maxwell equations across an infinitesimal pillbox or loop gives the dielectric-interface boundary conditionsandThus tangential $E$ and $H$ are continuous, as are normal $D$ and $B$.

Because $D=ϵE$, the normal electric field generally jumps according toThe tangential displacement field generally jumps because ${ϵ}_{+}\ne {ϵ}_{-}$. Since the permeability is common, continuity of tangential $H$ is continuity of tangential $B$; together with normal-$B$ continuity, every component of $B$ is continuous. The same common $\mu$ also makes the normal component of $H$ continuous.