Two-Dimensional Child-Langmuir Law

By considering uniform emission of electrons over a finite strip of width ${\mathit{W}\mathrm{in}\mathrm{a}\mathrm{planar}\mathrm{gap}\mathrm{of}\mathrm{gap}\mathrm{separation}\mathit{D},\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{e}-\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}-\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{m}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{w}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{w}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}.\mathrm{The}\mathrm{limiting}\mathrm{current}\mathrm{density}\mathrm{in}\mathrm{two}\mathrm{dimensions}\mathit{J}}_{\mathrm{CL}}$(2) in units of the classical one-dimensional value ${\mathit{J}}_{\mathrm{CL}}$(1) is found to be a monotonically decreasing function of W/D. More surprisingly, it is virtually independent of the external magnetic field that is imposed along the mean flow. These results were obtained from two different simulation codes, OOPIC and MAGIC.