Three-Dimensional Space Charge Flow

Abstract

The general theory of stationary beams of prescribed shape is developed taking into account the electrostatic interaction between the particles. The Hamilton‐Jacobi equation, expressed in a particular system of coordinates, allows the determination of the potential distribution inside the beam. Compatibility conditions for the equations are obtained, as well as a general rule for the determination of the feasibility of arbitrarily chosen trajectories. A theoretical expression results for the proportionality coefficient k in Child's law (J=kV) which relates current density J with the potential difference V between electrodes. The theory was applied to the particular case of a plane beam with hyperbolic boundaries. The beam was experimentally reproduced, providing values for k in close agreement with the theoretical ones.