Return current induced by injection of a relativistic electron beam into a plasma

The return current induced by a relativistic electron beam penetrating a plasma is studied by a macroscopic treatment. The beam pulse is described by a sine function of half period. The plasma current is governed by Ohm's law. An expression of the return current is derived. On the condition when 4 σ τ is much greater than 1, the return current builds up quickly and is large enough to compensate the beam current. The ratio of beam energy loss by Ohmic heating of the return current to that by Coulomb collisions is calculated, showing that the Ohmic heating of the return current is predominant