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

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.

When $|k{v}_z/\omega |\ll 1$ over the populated velocities, expand the denominator in powers of $k{v}_z/\omega$. Odd Gaussian moments vanish. The first two nonzero terms use $⟨v_z^2⟩=v_{th}^2$ and $⟨v_z^4⟩=3v_{th}^4$, yielding $1-{\omega }_p^2/{\omega }^2-3{\omega }_p^2{k}^2{v}_{th}^2/{\omega }^4+\cdots =0$. Solving consistently to order $k^2$ givesThis is a long-wavelength/high-phase-speed approximation, not an exact Maxwellian dispersion relation at arbitrary $k$.

Differentiate the approximate dispersion: $v_g=3k{v}_{th}^2/\omega$, while $v_{\varphi }=\omega /k$. Therefore $\boxed{v_g{v}_{\varphi }=3v_{th}^2}$. In the high-phase-speed regime that justified the approximation, the group speed is correspondingly small. Trying to make $v_g$ much larger than $v_{th}$ would put the phase speed into the thermally populated distribution, where resonant Landau damping is strong and the undamped-fluid estimate becomes unreliable. This explains the stated tendency for these electron electrostatic waves, not a universal prohibition on fast electromagnetic waves in plasma.