Two-dimensional space charged limiting current density of a long-pulse electron flow in drift space

Summary form only given. Formulations for the weakly relativistic two-dimensional electrostatic space charged limiting current density in drift space have been derived by considering the uniform injection of electrons over a finite strip of width W in a planar drift space of gap separation D. The two-dimensional limiting current density in units of the one-dimensional current density was found to monotonically increase with D/W. Under the approximation of W>>D in classical regime, the ratio of the limiting current density reduces to a simpler form: J(2D)/J(1D) ~ [1-(7/8jc)(D/W)]-1. When taking the emitter width to infinite (W→∞), the previous formula recovers to one-dimensional theory
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. These results are similar to W. B. Bridges' theory
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in classical regime. In simulations, The over-injection method and potential-matching method were used in the electrostatic particle-in-cell simulations to verify the two-dimensional theory, and the discrepancy between the simulation results the theoretical analysis is less than 5%.