Generating Function Method for Calculating the Potentials of Inhomogeneous Polyhedra

Alexey Vladimirovich Nenashev

doi:10.3389/fphy.2021.795693

Abstract

We propose a method of constructing analytical, closed-form expressions for electrostatic/Newtonian potentials of non-uniform polyhedral bodies, in which the density distributions are polynomials of coordinates. Possible applications of the proposed method are spread from astronomy to nanotechnology. The method is based on the use of the generating function for the potential. Explicit expressions for the potential are derived in the case of quadratic or cubic coordinate dependence of the density within a polyhedral body.

Highlights

Many problems in various physical applications are reduced to expressions of the form φ(R) ρ(r) |r − R| d3r. (1) VOne of them is the determination of the Coulomb potential from a given distribution of charge
The results of the present paper provide an insight into analytic properties of solutions of linear inhomogeneous partial differential equations (PDEs) possessing a geometry of a polyhedron
One can interpret Le(R) as a potential at point R of edge e considered as a uniform massive rod with the unit linear mass density. In these notations, generating function G(R, k) acquires the following form derived in Supplementary Section S2: FIGURE 2 | Unit vectors and radius-vectors associated with faces, edges, and vertices of the polyhedron

Summary

INTRODUCTION

Many problems in various physical applications are reduced to expressions of the form φ(R). If ρ(R) is a polynomial, the r.h.s. of Eq 6 contains only a finite number of terms This equation gives an analytical, closed-form expression for the Newtonian potential φ(R) for any polynomial density distribution ρ(R) within a polyhedral body, provided that coefficients G(0)(R), etc. Eq 8, along with the results of Section 2, provides a method of constructing analytical, closed-form expressions for Newtonian (or electrostatic) potentials of non-uniform polyhedral bodies, in which the density distributions are polynomials of coordinates. One can interpret Le(R) as a potential at point R of edge e considered as a uniform massive rod with the unit linear mass density In these notations, generating function G(R, k) acquires the following form derived in Supplementary Section S2: FIGURE 2 | Unit vectors and radius-vectors associated with faces, edges, and vertices of the polyhedron.

SERIES EXPANSION OF THE GENERATING FUNCTION

CONCLUSION