The Theory of the Pierce Type Electron Gun†

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Abstract
The problem of determining the Pierce electrodes in a gun reduces to a Cauchy problem on Laplace's equation. The questions of existence, uniqueness and instability of solutions to general problems of this type are considered, together with their relevance to the design of electron guns. A general procedure for solving a Cauchy problem is developed, and simplified for two-dimensional cases. This method is then used to determine the electrodes winch will maintain strip, wedge, cylindrical and conical-shaped beams. The Cauchy conditions for these problems are given by tho spaco-chargo limited potential from the appropriate complete diode solution.