Consider plane wave disturbances propagating along the magnetostatic field B 0 in a hot electron gas, whose
Question:
Consider plane wave disturbances propagating along the magnetostatic field B0 in a hot electron gas, whose equilibrium distribution function is homogeneous and isotropic. In spherical coordinates in velocity space (v, θ, ϕ) with B0 = B0ẑ and k along B0, as illustrated in Fig. 6, show that the linearized Vlasov equation reduces to
Verify that this differential equation has the formal solution
where v' is the velocity vector with components ( v, θ, ϕ'). Note that
Perform the integral in this expression for f1(v) to obtain
where A= –(w – k · v)/Ωce· From Maxwell equations obtain the relation
where Et = E – Ezẑ is the transverse part of the electric field E. Using the expression for f1(v) in this equation, show that we obtain a dispersion relation with three wave solutions, which are the usual Landau damped longitudinal waves and the left and the right circularly polarized waves (with Ex = ±iEy)·
Figure 6.
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