This article presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables (i.e., variables that follow particle paths) the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity v 0, electric field E 0, and current J 0. For steady solutions, this approach clarifies the origin of the maximal current J max, derived by Child and Langmuir for v 0=0 and by Jaffe for v 0>0. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux J 0>J max, which leads to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J max.
|Original language||English (US)|
|Journal||Physical Review E - Statistical, Nonlinear, and Soft Matter Physics|
|State||Published - May 18 2012|
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics