Sheath Equivalent Electrical Circuit Model for Equilibrium Child Sheath Description

Abstract

The Child–Langmuir law is well known in the fields of electronics and plasma physics. This has relevance to the analytical description of the space charge sheath limited current and its dependence on the space charge sheath width. For known values of the applied wall voltage, the current and width are two unknowns to be determined self-consistently and independently. The Child–Langmuir law alone is not sufficient for self-consistent description of the equilibrium Child sheath. Here we use the concept of sheath equivalent electrical circuit model and propose an idea of current equivalence principle to develop an additional equation which closes the usual Child sheath equation. Thereafter, we carry out self-consistent and independent determination of the Child sheath current and width. It is found that the steady state (time averaged) view of Kirchoff’s law fulfils the closure property of the Child sheath equation and verifies the constant current approximation of conventional Child sheath description. The proposed idea of current equivalence principle states that the hydrodynamic and electrical circuit views of the Child sheath current are equivalent, which is supposed to govern the equilibrium and dynamic behaviors of the Child sheath.