The zero gap conjecture

Abstract

The conjecture asserts that one electron optical device modeled having no gaps and another in which there are arbitrarily small gaps between the contacting elements are equivalent. It is verified using as an example a high resolution cylindrical mirror analyzer having contacting elements at different potentials. The verification proceeds by showing the equivalence between an air gap separating the elements and that of a small segment separating the elements at voltage V. Next use is made of this small segment at ever decreasing lengths in order to complete the assertion. Lastly the net charge located in the vicinity of the contact point is numerically calculated and it is found that this net charge becomes vanishingly small as the mesh point density surrounding the contact point becomes increasingly large. Thus there is no net charge in the contact region that can affect trajectories, a result consistent with the conjecture itself.