Analytic expression for the electric potential in the plasma sheath

An expression for the spatial dependence of the electric potential in a collisionless and source-free planar plasma sheath is presented. This expression is derived by analogy with Child's law and approaches Child's law asymptotically as the potential drop phi /sub W/ across the sheath becomes large, mod e phi /sub W//kT/sub e/ mod >10/sup 4/. Here k is Boltzmann's constant, T/sub e/ is the electron temperature, and e is the electronic charge. Comparison with numerical solutions of the model equations indicates that the sheath thickness and potential variation predicted by this improved Child's law are accurate for mod e phi /sub W//kT/sub e/ mod >10. In contrast, the authors find that Child's law is accurate only when mod e phi /sub W//kT/sub e/ mod >10/sup 4/.<
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