Electrode shapes for a cylindrical electron beam

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Abstract

The problem of obtaining the electrode shapes which will maintain a cylindrical beam of space-charge limited current reduces to the problem of solving Laplace's equation for the electrostatic potential outside the space-charge region, with the appropriate boundary conditions on the surface of the beam. A new approximate analytical solution of the Laplace equation has been obtained, and numerical tables have been developed for designing the electrode shapes. A plot of the electrode shapes, which is applicable to either hollow or solid beams, has been drawn up. The shape of the outside electrodes is the same for either type of beam, but the hollow beam requires an additional set of electrodes located inside the beam. The shape of the outside electrodes agrees fairly well with that obtained previously by Pierce,(1) Harrison,(2) and Kuo Chu Ho and Moon.(3)