Longitudinal oscillations in a boundary neutralized electron beam with a boundary

Longitudinal oscillation of a neutralized electron beam with a boundary in a strong axial magnetic field is discussed. The beam is assumed to be cylindrical and to be confined in a conducting cylinder, or to be planar and to be held between two conducting plates. A sufficient and, in some cases, approximately necessary condition for stability is obtained. To be stable, the beam current should be below a certain value which depends on the electron velocity, the ratio of electron to ion mass and the geometrical dimensions of the beam and the conductor. The validity of the approximation is also studied.