A new approach to the Child-Langmuir law

TL;DR

This paper introduces a novel analytical method to determine the space charge limited current in vacuum tube diodes, providing new physical insights into the Child-Langmuir law without solving complex nonlinear equations.

Contribution

It presents a new variable-based approach to analyze charged particle motion, simplifying the derivation of the Child-Langmuir law and extending it to cases with initial velocity.

Findings

Derived the space charge limited current for non-relativistic particles with zero initial velocity

Extended the analysis to particles with non-zero initial velocity

Provided a new physical interpretation of the Child-Langmuir law

Abstract

We analyze the motion of charged particles in a vacuum tube diode using a new set of variables. We obtain the space charge limited current for a charged particle moving non-relativistically in one dimension for the case of zero and non zero initial velocity. Our approach gives a new physical insight into the Child-Langmuir law and avoids the need of solving a nonlinear differential equation.