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# Space charge flow in spherical coordinates with applications to magnetron injection guns

## Abstract

In a recent publication, Kirstein [1] called attention to the possibility of calculating space charge flow by separation of variables in spherical coordinates (r, θ, φ). To be of practical value, detailed information about the solutions must be obtained. It appears that closed form solutions for the equations of motion are not practical even if they are possible, since one needs the numerical data for the solutions. It is usually more convenient to have a computer solve the differential equations than to evaluate a closed form solution of the differential equations. All of the equations for computations will be written as first order differential equations with respect to time as the independent variable. This makes each of the necessary differential equations well behaved at the cathode and allows

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## References

- [1]KIRSTEIN, P.T.,
*Some Solutions to the Equations of Steady Space Charge Flow in Magnet ic Fields*, JEC Vol. VII, Nov. 59, pp. 417, 422.Google Scholar - [2]WATERS, W.E.,
*Azimuthal Electron Flow in a Spherical Diode*, JAP, 30, March 1959, pp. 368–373.Google Scholar - [3]Personal Communication from D.C. Colburn at Stanford University, August 1960.Google Scholar
- [4]HARKER, K.J.,
*Determination of Electrode Shapes for Axially Symmetric Electron Guns*, JAP, 31, Dec. 1960, p. 2165.MathSciNetGoogle Scholar - [5]WIGNER, E.P.,
*Gruppentheorie Und Ihre Anwendung Auf Die Quant ennechanik Der Atomspektren*, Braunschweig f. Vieweg und Sohn, 1931.Google Scholar - [6]KINO, G.S. and TAYLOR, N.J.,
*The Design and Performance of a Magnetron-Injection Gun*, IRE Trans, on Electron Devices, Vol. ED-9, pp.1-11, January, 1962.CrossRefGoogle Scholar