Electric field distribution on knife-edge field emitters

Conformal mapping is used to calculate the electric field on a knife-edge cathode, modeled as a rectangular ridge on a flat surface. It is found that the field enhancement factor scales approximately as the square root of the height-to-width ratio of the knife edge. A simple analytic approximation for the divergent electric field in the immediate vicinity of the sharp edge is derived. When a smaller knife edge is placed on top of a larger one, both assumed to have large height-to-width ratios, the composite field enhancement factor is shown to be approximately equal to the product of the field enhancement factor of the individual knife edges, thereby proving the conjecture on multiplication of field enhancement factors for one special case.