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

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.