Triangular Electrostatic Ion Traps Revisited

Motivated by a mathematical model of triangular electrostatic ion trap developed in our previous papers, we present a few analogous results for three uniformly charged coplanar discs. To this end we elaborate upon several earlier results on the Coulomb potential and equilibrium points of three non-collinear point charges. In particular, we establish that the Coulomb potential of the so-called trapping charges is a Morse function and give an explicit formula for its hessian at the trapped point. These facts combined with the recent results of Y.-L. Tsai on three point charges with equal magnitudes, enable us to establish that the Coulomb potential of sufficiently small identical uniformly charged discs centered at the vertices of regular triangle has exactly seven critical points. This implies that, for triangular shapes sufficiently close to the regular triangle, the number of equilibrium points of appropriately charged small discs centered at the vertices is not less than seven.