Space-charge-limited 2D electron flow between two flat electrodes in a strong magnetic field

An approximate analytic solution is constructed for the 2D space-charge-limited emission by a cathode surrounded by nonemitting conducting ledges of width Lambda. An essentially exact solution (via conformal mapping) of the electrostatic problem in vacuum is matched to the solution of a linearized problem in the space charge region whose boundaries are sharp due to the presence of a strong magnetic field. The current density growth in a narrow interval near the edges of the cathode depends strongly on Lambda. We obtain an empirical formula for the total current as a function of Lambda which extends to more general cathode geometries.