| Abstract: | An analytic solution of the equations of a regular electrostatic beam in the presence of emission from an arbitrary surface under total space charge conditions is given. It is assumed that the emitter is a coordinate surface x1= const in an orthogonal system xi (i=1, 2, 3), and the emission-current density J is a given function J(x2, x3). The solution is represented in the form of series in x with coefficients that are functions of x2, x3 and determined from recurrence relations. In expansion along the length of an arc of the curvilinear axis x1, which is orthogonal to the emitter, the first correction of the Child-Langmuir 3/2 law is determined only by the total curvature (the sum of the principal curvatures) of the emitting surface. Solution of the problem in the formulation in question permits determination of the collector shape that ensures the given distribution of the emissioncurrent density over the given surface. |
