Similarity of limiting currents in planar and cylindrical crossed- eld diodes

Limiting currents in planar crossedeld diodes have been studied for over 70 years [1][3] for B < BHull and more recently for B > BH [4]. Recent simulations of cylindrical crossedeld diodes for anode/cathode radius ratios of 2 and 5 indicate that the limiting current curve in cylindrical diodes follows the planar theory and simulations very closely [5]. Newer simulations for larger radius ratios (10, 20) show somewhat larger limits. A possible explanation for this behavior is also examined. 1 Background 1.1 Space Charge Limited Current in Cylindrical Systems For zero applied magnetic eld, we need to solve the one dimensional Poisson equation in cylindrical coordinates 1 r @ @r (r @V (r) @r ) = (r) (1) where V (r) is the local potential and (r) is the local charge density. The kinetic energy gained by a charged particle of mass m, charge e and velocity v, accelerated through a time-independent potential drop V is given by (particle starting at V = 0 = v) 1 2 mv = eV: (2) The current density J(r) inside the system can be written in terms of the charge density and velocity as J(r) = (r)v(r): (3)