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

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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.