Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S16 / 3 / 1 / Solution

Take electrostatic perturbations proportional to $e^{i(kz-\omega t)}$, with ions immobile on the fast electron time scale. Linearized Vlasov gives $f_1=-i(q{E}_1/m)(\partial f_0/\partial v_z)/(\omega -k{v}_z)$. Integrating over transverse velocities gives $F_0(v_z)=n_0\sqrt{m/(2\pi k_{B}T)}{e}^{-m{v}_z^2/(2k_{B}T)}$. Combine the perturbed charge density with Poisson's equation:Since $F_0^{\prime }=-m{v}_z{F}_0/(k_{B}T)$, this is preciselyThe pole selects particles moving at phase speed. A causal initial-value prescription specifies how the contour passes it and produces resonant energy exchange, including Landau damping for a decreasing Maxwellian. A bare real improper integral without that prescription is incomplete. The unclosed brace in the printed Maxwellian exponent is typographical; the normalized Gaussian above fixes its meaning.