## Abstract

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) |
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Article number | 056408 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 85 |

Issue number | 5 |

DOIs | |

State | Published - May 18 2012 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics