Weak magnetic-field expansion for the self-consistent steady-state potential in a planar crossed-field diode

In a planar crossed-field diode, the steady-state flow in the space-charge-limited régime is governed by a nonlinear integrodifferential equation. We have derived a generalized Fry-Langrnuir equation for the potential which is self-consistent to second order in the normalized magnetic field µ. The small-µ crossed-field potential may be considered as the Fry-Langmuir potential of the corresponding unmagnetized diode, i.e., with the same diode parameters but with zero magnetic field, plus the steady-state second-order perturbation produced by the magnetic field. Owing to the symmetry of the unbounded, planar geometry, there is no first-order perturbation. From four universal functions (the Fry-Langrnuir function and three additional functions we have derived), one can determine the potential and density profiles of any space-charge-limited planar crossed-field diode for which µ is small. It is shown that the crossed-field potential minimum is deepened and shifted toward the anode with respect to the unmagnetized potential minimum.