A simple physical derivation of Child–Langmuir space-charge-limited emission using vacuum capacitance

The fundamental Child–Langmuir limit on the maximum current density in a vacuum between two infinite parallel electrodes is one of the most well known and often applied rules of plasma physics. We develop a simple model using vacuum capacitance, conservation of energy, and conservation of charge to derive the Child–Langmuir space-charge-limited emission. This capacitive model provides physical insight into the origins of the well known (voltage)3/2/(gap distance)2 scaling of the classical current density and does not require the solution of the nonlinear differential equation normally associated with the Child–Langmuir formulation. In addition, the full spacecharge-limited solution is reproduced without imposing the condition that the electric field be driven to zero at the cathode surface.