Exact Solution of Poisson's Equation for Space-Charge-Limited Flow in a Relativistic Planar Diode

Abstract

Poisson's equation, governing space-charge-limited flow in a relativistic planar diode, is solved assuming the initial velocities of the accelerated particles are zero, through the use of two power series convergent in the potential range 0≤V≤2m0c2/Ze and 2m0c2/Ze≤V < ∞. In the region of lower potential the solution is expressed in a power series in U, a normalized potential. As U becomes small the solution reduces to the well-known Child's Law. In the region of higher potential, a power series in inverse powers of U is employed. As U becomes large the solution reduces to the ultra-relativistic form obtained if v, the particle velocity, can be considered equal to the speed of light. Convergence of both series is rapid, and it is only necessary to retain a few terms to realize a high degree of accuracy.