Space-charge limited current of hollow relativistic electron beam in coaxial waveguide

An exact analytical expression for the space-charge-limited current of hollow relativistic electron beam in a coaxial waveguide is derived and compared with the existing formula for extremely thin hollow electron beam and that in cylindrical waveguide. Also, the numerical result from nonlinear partial differential equation of potential for the space-charge-limited current of hollow relativistic electron beam in a coaxial wave-guide is given and is much larger than that from the analytical expression. Hollow electron beam has higher space-charge limited current in coaxial waveguide than that in cylindrical waveguide and it higher when electron beam approaches the conductors of waveguide. This theoretical analysis can be used to guide the design of high power microwave coaxial waveguide.