The approximate calculation of 2D field strength with a Coulomb potential by using multipole expansion

Abstract

The necessary formulae are derived for approximate computation of the Coulomb-type potential and the field strength, generated by system of particles in 2D case. On the one hand, these formulae represent the multipole expansion in which monopole, dipole, quadrupole and octupole terms are taken into account that allows for computing the potential and the field strength, generated by set of the charges with rather high accuracy. On the other hand, polynomial expansions of the obtained formulae are derived up to the cubic terms with respect to the observation point coordinates, which make it possible to compute the field strength at closely located points approximately. The expressions are derived that permit one to compute the multipole moments for the system of charges, which consists of smaller subsystems. Such formulae are required in tree-codes when the hierarchical (tree-type) structure of geometrical regions is introduced in the region of the initial system of the charges. In particular case these expressions allows for changing of the reference point of multipole moments computation (multipole moments shifting). The test problem is considered for rather small number of the charges, which shows the dependency of the relative error of the field strength reconstruction on the type of multipole expansion and number of considered terms in the Taylor expansion.