Three-dimensional solution to the problem concerning the potential of electron bunches in crossed fields

Three-dimensional expressions are obtained for the potential of electron bunches of arbitrary configuration in crossed fields in an M-type device with allowance for the conducting planes of the electrodes. Solutions to the three-dimensional Poisson equation are obtained along with conditions of potential periodicity, physically equivalent to the assumption of a sequence of identical electron bunches. The three-dimensional solutions are compared with solutions averaged with respect to the Z axis as well as with two-dimensional solutions.