Current and current density of a finite-width, space-charge-limited electron beam in two-dimensional, parallel-plate geometry

The current carried by a steady-state, finite-width beam produced by space-charge-limited emission in two-dimensional parallel plate geometry is known to diverge strongly from estimates based on the classic one-dimensional Child–Langmuir problem. The two-dimensional problem presents formidable analytic difficulties, but a numerical approach to this problem has been developed. The approach simultaneously determines the electrostatic potential and the unknown current density profile. Calculations show that the total current is a function of the dimensionless ratio of beam width (w) to anode–cathode gap (d), but that the current density profile varies with both w and d separately.