A weak electron beam of density streams at through a background Maxwellian electron…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Kinetic theory, collisions and transport · Kinetic theory, collisions and transport

Plasma Physics, Ch Hot plasmas and free energy: beam instabilities, cyclotron damping and Bernstein waves

Problem

A weak electron beam of density

n_b = 2\times10^{12}\,\mathrm{m^{-3}}

streams at

v_b = 3\times10^{7}\,\mathrm{m\,s^{-1}}

through a background Maxwellian electron plasma of density

n_e = 10^{16}\,\mathrm{m^{-3}}

and temperature

T_e = 50\,\mathrm{eV}

, so that

v_{te} \ll v_b

. The combined distribution

f_0(v)

(bulk plus beam) has a single local minimum between the bulk peak and the beam peak. ((a)) Predict, from the sign of

\hat f_0{}'(v)

on each side of that minimum, over what range of phase velocities

\omega/k

waves grow. ((b)) The beam subsequently relaxes to a flat plateau extending from near zero velocity up to

v_b

, with the number of beam electrons conserved. Predict the kinetic energy per unit volume that is released to the waves during this relaxation.

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