Abstract
Solutions of Poisson's equation for a space-charge-limited spherical diode, in which electron trajectories are great circles rather than radial lines, are developed. It is shown how Poisson's equation may be separated to yield ordinary differential equations. The radial equation is solved analytically, whereas accurate numerical solutions of the azimuthal equation are given. A method of truncating the flow so as to produce a ``bowl-shaped'' electron gun, leading eventually to an annular electron beam, is presented, along with power-series solutions of Laplace's equation both inside and outside the region of the flow. The latter series are useful for calculating the shapes of anodes and focusing electrodes.
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