  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.
