  In this paper we analyze the motion of charged particles in a vacuum tube
diode by solving linear differential equations. Our analysis is based on
expressing the volume charge density as a function of the current density and
coordinates only, while in the usual scheme the volume charge density is
expressed as a function of the current density and electrostatic potential. Our
approach gives the well known behavior of the classical current density
proportional to the three-halves power of the bias potential and inversely
proportional to the square of the gap distance between the electrodes, and does
not require the solution of the nonlinear differential equation normally
associated with the Child-Langmuir formulation.
