In problem 19.4 suppose that in the equilibrium state the velocity distribution function of the electrons is

Question:

In problem 19.4 suppose that in the equilibrium state the velocity distribution function of the electrons is given by

which corresponds to an isotropic distribution but with the electrons drifting with macroscopic speed u₀ along B₀. Show that, with this choice of f₀(v), the dispersion relation for the right circularly polarized wave reduces to

For the limiting case of T_e = 0, find the form of the distribution function f₀(v) and show that the dispersion relation reduces to

Data from problem 19.4.

Consider an electron gas immersed in a uniform magnetostatic field B₀ and characterized by the following modified Maxwellian distribution function:

Use this distribution function in the dispersion relation for the right circularly polarized transverse wave propagating along B₀, given in (2.69), and evaluate the integrals to obtain the following dispersion relation

where

Analyze this dispersion relation to verify the existence or not of instabilities (positive imaginary part of ω) and/or damping (negative imaginary part of ω) of the wave amplitude, considering the propagation coefficient k = kẑ to be real. Determine the cyclotron damping coefficient. Analyze also the results considering the isotropic case for which T_∥ = T_⊥.

Equation 2.69.