Multi-dimensional child-langmuir law for a crossed-field gap

Summary form only given. The classical space-charge limited current or the Child-Langmuir (CL) law continues to be an active field of research till today. The 1D classical CL law is only extended to the multi-dimensional model, which is valid in the 2 classical, weakly-relativistic and quantum regime, recently. For crossed-field devices, where a transverse magnetic field is applied across the vacuum gap, the 1D CL law for both the laminar and turbulent region were reported more than a decade ago. In this paper, we will present an analytical formulation to calculate the maximum CL current density emitted from a finite 2D emission area, such as a 2D infinite strip, for a crossed-field gap. The analytical results are then compared and verified with particle-in-cell simulations. The implication and the application of this model will be discussed