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Two-dimensional axisymmetric Child-Langmuir scaling law

The one-dimensional Child-Langmuir law has been extended to two dimensions by numerical simulation in planar geometries . The space charge limit in two dimensions is examined. Simulations were done with no applied magnetic field as well as with a large longitudinal magnetic field to restrict motion of particles to 1D. The ratio of the observed current density limit JCL2 to the theoretical one-dimensional value JCL1 is found to be a mono- tonically decreasing function of the ratio of emission radius (r) to gap separation (L). This result is in agreement with the planar results, where the emission area is proportional to the cathode width. The simulations were run in the particle in cell code. Least-squares fits to a log-log plot for no applied magnetic field.