Space charge flow in spherical coordinates with applications to magnetron injection guns
In a recent publication, Kirstein  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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