Physical explanation of the universal "(separation) –3 " law found numerically for the electrostatic interaction between two protruding nanostructures Richard G Forbes Advanced Technology Institute & Department of Electrical and Electronic Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK E-mail: r.forbes@trinity.cantab.net Received Abstract Two conducting nanostructures situated on a conducting base-plate, and subject to a common externally applied macroscopic electrostatic field, interact because their electrons are part of a common electron-thermodynamic system. Except at very small separations, the interaction reduces the apex field enhancement factor (FEF) of each nanostructure, by means of an effect that has been called "charge blunting". A parameter of interest is the fractional reduction (–δ) of the apex FEF, as compared with the apex FEF for the same emitter when standing alone on the base-plate. For systems composed of two or a few identical post-like emitters, or regular arrays of such emitters, interaction details have been investigated both by methods based on numerical solution of Laplace's equation, and by the use of line-charge models. For post separations c comparable with the post height h, several authors have shown that the variation of (–δ) 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 sufficiently large c-values, (–δ) falls off as c –3 . Numerical Laplace-type simulations carried out by de Assis and Dall'Agnol (arΧiv1711.00601v2) have confirmed this limiting c –3 dependence for six different situations involving pairs of protruding nanostructures, and have led them to suggest that it may be 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 c –3 dependence, for large c. An implication is that the quasi- exponential formula found applicable for c~h is simply a good fitting formula, particularly in this range. A second implication is that the FSEPP model, which currently is used mainly in nanoscience, may have much wider applicability to electrostatic phenomena.