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 φ_W across the sheath becomes large, |eφ_W/kT_e|>10⁴. Here k is Boltzmann's constant, T_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 |eφ_W/kT_e|>10. In contrast, the authors find that Child's law is accurate only when |e φ_W/kT_e|>10⁴