Space charge limited current of beam in radial transmission line

In order to study the space charge limited current and its related physical quantities, considering the infinitely long radial direction and the current density being constant in axial direction, the Poisson equation for electrical potential is solved theoretically and the expression of the space charge limited current of the beam is obtained. Then the non-linear Poisson equation which takes into consideration the non-uniform distribution of the charge density in axial direction is solved numerically. The numerical results and analytical results are compared. The results show that the radial beam has a higher space charge limited current in radial transmission line. When the beam value approaches to the space charge limited current, the non-uniform distribution of the charge density increases significantly. The relevant conclusion can provide guidance on the design of radial high power microwave devices.