Velocity shear driven instabilities of an unneutralized electron beam

The stability of a planar electron beam with a linear velocity shear both parallel and perpendicular to a fixed, uniform magnetic field is examined. In the case of a beam bounded by conducting walls, an exact criterion for stability is determined for perturbations of arbitrary wavelength. A similar, although more involved, criterion is given for a beam isolated from conducting walls. Estimates of the growth rate of the instabilities are determined for both long and short wavelength perturbations.