Abstract
For the vicinities of arbitrary axially symmetric emitting surfaces, solutions to the beam equation are obtained in the case of counter motion of nonrelativistic nonmagnetized ions and relativistic electrons in an external magnetic field. The boundary value problem of determining the emission current in the ρ mode and the electric field on the cathode in the T mode or the problem of determining the shape of the plasma boundary in the r mode, which is related to the enhancement of the accuracy of trajectory analysis programs, is considered.
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