Consider a uniform

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

Plasma Physics, Ch Plasma oscillations and light in an unmagnetised plasma

Problem

Consider a uniform, unmagnetised, cold, collisionless plasma of electrons (density nen_e, mass mem_e, charge −e-e) with fixed neutralising ions, perturbed by a plane wave E(x,t)=E0ei(k⋅x−ωt)\mathbf E(\mathbf x,t)=\mathbf E_0e^{i(\mathbf k\cdot\mathbf x-\omega t)}. Ion motion, collisions and pressure are neglected throughout. ((a)) From the linearised electron momentum equation, find the electron velocity perturbation v1\mathbf v_1 in terms of E0\mathbf E_0, form the current j1=−enev1\mathbf j_1=-en_e\mathbf v_1, and hence write the conductivity σ(ω)\sigma(\omega) and the dielectric function ϵ(ω)=1+iσ(ω)/ϵ0ω\epsilon(\omega)=1+i\sigma(\omega)/\epsilon_0\omega. ((b)) From Maxwell's equations for the same mode, eliminate B\mathbf B between Faraday's law and the Ampere-Maxwell law to obtain n×(n×E)+ϵ⋅E=0\mathbf n\times(\mathbf n\times\mathbf E)+\boldsymbol\epsilon\cdot\mathbf E=0 with n=ck/ω\mathbf n=c\mathbf k/\omega. For the isotropic ϵ=ϵ(ω)I\boldsymbol\epsilon=\epsilon(\omega)\mathsf I found in part (a), show that a nonzero E\mathbf E requires either E∥k\mathbf E\parallel\mathbf k with ϵ(ω)=0\epsilon(\omega)=0, or E⊥k\mathbf E\perp\mathbf k with n2=ϵ(ω)n^2=\epsilon(\omega). ((c)) Confirm that both branches reduce to the correct vacuum results in the limit ne→0n_e\to0.

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