Space charge limited flow of a thin electron beam confined by a strong   magnetic field
  An approximate analytic theory is developed and implemented numerically for calculating the space charge limited current and electric field of a thin cylindrical beam or current sheet between two wide parallel electrodes. The flow is confined by a sufficiently strong magnetic field. Assuming that the potential and current density are almost homogeneous in the direction transversal to the flow we compute the beam current and profile by a variational method. We find that the average current density scales as the reciprocal of the beam width when the latter becomes very small. The total cylindrical beam current thus decreases proportionly to its diameter while the total current of a sheet becomes almost independent of the width in this regime. 
A.Rokhlenko and J. L. Lebowitz