Steady-state configuration of a relativistic electron beam in a semiinfinite vacuum waveguide

Simple approximate equations are derived for the steady-state configurations of the electrostatic potential and density of a relativistic electron beam injected across a metal foil into a semiinfinite vacuum waveguide. The beam currents here are less than or equal to the maximum current for an infinitesimally thick electron beam in an unbounded vacuum diode. It is assumed that the beam is of infinitesimal thickness and that the magnetic guide field is infinitely strong. The foil and the waveguide walls are at zero potential. It is also assumed that all the variables depend only on r and z, that is, that the problem is axisymmetric.