A uniform, unmagnetised plasma has electrons of charge , mass , equilibrium density ,…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Plasma Physics, Ch Waves that need pressure or tension: Langmuir, ion-acoustic, Alfven and magnetosonic

Problem

A uniform, unmagnetised plasma has electrons of charge −e-e, mass mem_e, equilibrium density n0n_0, temperature TeT_e, moving against a fixed, uniform neutralising ion background. A small one-dimensional electrostatic perturbation varies as ei(kx−ωt)e^{i(kx-\omega t)}. ((a)) Starting from the linearised electron continuity equation, the linearised electron momentum equation with pressure closure pe∝neγep_e \propto n_e^{\gamma_e}, and Poisson's equation, derive ω2=ωpe2+γek2vte2,vte2=Te/me.\omega^2 = \omega_{pe}^2 + \gamma_e k^2 v_{te}^2, \qquad v_{te}^2 = T_e/m_e. ((b)) State the rule (from the fluid closures) that fixes γe=3\gamma_e = 3 for this wave, and write the resulting Bohm-Gross relation. ((c)) For ne=1.0×1018 m−3n_e = 1.0\times10^{18}\,\mathrm{m^{-3}}, Te=5.0 eVT_e = 5.0\,\mathrm{eV} and a wave with kλDe=0.40k\lambda_{De} = 0.40, evaluate ω\omega, the phase velocity ω/k\omega/k and the group velocity vg=3kvte2/ωv_g = 3kv_{te}^2/\omega.

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