Simple theory for the two-dimensional Child-Langmuir law

This paper presents, for the first time, a simple analytic theory for the two-dimensional (2D) Child-Langmuir law. For electron emission over a finite patch on a planar cathode, the limiting current density is derived approximately from first principles. The scaling laws are in excellent agreement with simulation results. They predict the onset of virtual cathode formation in a 2D geometry; they also indicate that electrons emitted from a cathode over only a restricted area may have a current density much exceeding the classical (1D) Child-Langmuir value.