Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 36B (Electrodynamics)

The source-free Maxwell equations in a linear medium arewith $D=ϵE$ and $B=\mu H$.

Take complex plane wavesThe divergence equations giveand Faraday's law givesAmpère's law givesSubstituting the expression for $B_0$ and using
$k\times (k\times E_0)=-|k{|}^2{E}_0$ yieldsHence the plane electromagnetic wave in a linear medium hasand equivalently

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.

Choose the interface as $x=0$ and write the incident, reflected, and transmitted fields as plane waves with wavevectors $k_I,k_R,k_T$. The boundary conditions must hold for every tangential coordinate and every time. The phase matching at a planar wave interface therefore requiresand equality of tangential wavevectors:

The incident and reflected waves are in the same medium, so
$|k_I|=|k_R|$. Equality of their tangential components giveshence

At fixed frequency, $|k_{\pm }|=\omega /v_{\pm }$ is proportional to the refractive index $n_{\pm }$. Tangential phase matching between incident and transmitted waves gives the Snell law for electromagnetic waves:orHere $n_{\pm }=c/v_{\pm }=c\sqrt{{ϵ}_{\pm }\mu }$ if the vacuum speed $c$ is used to normalize the index.

Take the plane of incidence to be the $xy$-plane, so all three electric polarizations point along $z$. Write their signed scalar amplitudes as $E_I,E_R,E_T$. Tangential-$E$ continuity givesFor each plane wave,Continuity of its tangential component giveswhere the common proportionality between $|k|$ and $n$ cancels. Solving the two equations gives the transverse-electric Fresnel reflection coefficientUsing Snell's law to eliminate ${\theta }_T$,Since ${ϵ}_{+}>{ϵ}_{-}$ and $\mu$ is common, $n_{+}>n_{-}$. ThereforeThe numerator is consequently strictly negative throughout
$0\le {\theta }_I\le \pi /2$, andThus transverse-electric polarization has no Brewster angle in this setting.