Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 1 / b / Solution

The Faraday law and Ampere-Maxwell law give $\delta \mathbf{B}=(c/\omega )\mathbf{k}\times \delta \mathbf{E}$ andWith refractive index $n=kc/\omega$, the linear system is thereforeSetHere $R$ labels the cold-plasma right-handed wave and $L$ the cold-plasma left-handed wave, with their polarizations fixed explicitly in d. In the electron-only response,Writing $N=n^2$, expansion of the determinant yields the dispersion relationEquivalently,Division givesThe undivided determinant equation remains valid when this rational form has simultaneous zero factors.