Electron Beams in Axially Symmetrical Electric and Magnetic Fields

The problem considered in this paper is the formulation of the equations governing the motion of an electron beam in axially symmetrical magnetic and electric fields. The equations are obtained for the trajectories of the electrons along the outer edge of the beam for the most general case, in which there are both axial and radial components of the fields. It is shown that, as a result of symmetry, the combined effects of the electric and the magnetic fields can be expressed as a single generalized potential function which depends only on the axial and radial space co-ordinates. This permits one to express the axial and radial force components as the axial and radial components of the gradient of this potential function. Numerical solutions have been obtained by numerical integration for the trajectories in a uniform magnetic field. Curves are presented in normalized form, giving the results of these solutions for cases likely to be encountered in practice. It is shown that there exists an equilibrium radius for which the net radial forces acting on the electrons is zero, and that the outer radius of the beam will oscillate about this equilibrium value, the amplitude being nonsymmetrical and depending upon the initial conditions, and the wavelength (distance between successive maxima) depending upon the amplitude.