Two-Dimensional Child-Langmuir Law

Abstract

By considering uniform emission of electrons over a finite strip of width ${\mathit{W}\phantom{\rule{0ex}{0ex}}\mathrm{in}\mathrm{a}\mathrm{planar}\mathrm{gap}\mathrm{of}\mathrm{gap}\mathrm{separation}\mathit{D},\phantom{\rule{0ex}{0ex}}\mathrm{w}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\phantom{\rule{0ex}{0ex}}\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\phantom{\rule{0ex}{0ex}}\mathrm{o}\mathrm{n}\mathrm{e}\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\phantom{\rule{0ex}{0ex}}\mathrm{C}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}\ensuremath{-}\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{m}\mathrm{u}\mathrm{i}\mathrm{r}\phantom{\rule{0ex}{0ex}}\mathrm{l}\mathrm{a}\mathrm{w}\phantom{\rule{0ex}{0ex}}\mathrm{t}\mathrm{o}\phantom{\rule{0ex}{0ex}}\mathrm{t}\mathrm{w}\mathrm{o}\phantom{\rule{0ex}{0ex}}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}.\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{limiting}\mathrm{current}\mathrm{density}\mathrm{in}\mathrm{two}\mathrm{dimensions}\mathit{J}}_{\mathrm{CL}}$(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.