Physical explanation of the universal "inverse-3rd-power-of-separation" law found numerically for the electrostatic interaction between two protruding nanostructures
Two conducting nanostructures on a conducting base-plate, and with a common applied electrostatic field, interact because their electrons are a common electron-thermodynamic system. Except at small separations, the interaction reduces the apex field enhancement factor (FEF) of each nanostructure, by means of "charge blunting". A parameter of interest is the fractional reduction (-d) of the apex FEF, as compared with the apex FEF for the same emitter when standing alone on the base-plate. For systems of two or a few identical post-like emitters, or regular arrays of such emitters, details have been investigated by methods based on numerical solution of Laplace's equation, and by using line-charge models. For post separations c comparable with post height h, several authors have shown that the variation of (-d) with c is well described by formulae having exponential or quasi-exponential form. By contrast, explorations of the two-emitter situation using the "floating-sphere-at-emitter-plane potential" (FSEPP) model have predicted that, for large c-values, (-d) falls off as 1/c*c*c. Numerical Laplace-type simulations carried out by de Assis and Dall'Agnol (arXiv:1711.00601v2) have confirmed this limiting dependence for six different situations involving pairs of protruding nanostructures; hence they suggest it as an universal law. By using the FSEPP model for the central structure, and by adopting a "first moments" representation for the distant structure, this letter shows that a clear physical reason can be given for this numerically discovered general limiting (1/c*c*c) dependence. An implication is that the quasi-exponential formula found useful for c comparable with h is simply a good fitting formula. A second implication is that the FSEPP model, which currently is used mainly in nanoscience, may have much wider applicability to electrostatic phenomena.