Shielded point charges and multipoles

Abstract

+ 530.1Complex electric potentials and energy relations for shielded point charges and point multipoles of any arbitrary orders are derived. It is demonstrated that both the complex electric potentials of shielded point multipoles and their interaction energies can be expressed through derivatives of the Green's complex function. As an example, relations for the complex potentials of point multipoles of all orders in a half-plane are presented. It is indicated that the mathematical apparatus of shielded point multipoles can be used, in particular, for successive refinement of estimates of the shielded conductor capacitance.In the present work, concepts of point multipoles in a plane are analyzed in detail. It is demonstrated that the electric field of a system of round parallel wires can be approximately expressed on the basis of the Gauss variational principle [1] with any required degree of accuracy through fields of point multipoles of individual wires [2]. In [3,4], electrostatics problems in a plane were considered, and it was demonstrated that the field of the approximating point charge is closely related to the concept of the internal conformal radius ( ) A z , so that the quantity analogous to the capacitance of the surface relative to a point specified in [5], namely, the straight line capacitance relative to point z can be determined as follows:where R is the normalization constant [3]. If we replace the electric field of the shielded conductor outside of it by the field of the shielded point charge located inside of the conductor at point z , the estimate from below of the conductor capacitance relative to the screen [3]will correspond to this approximation. Here s ( ) А z and ( ) п А z are the conformal radii of the screen space and conductor volume relative to the point z .A question naturally arises on the subsequent refinement of estimate (1). It can be refined by approximation of the electric field of the shielded conductor by fields of the shielded point multipoles located at one or several points inside of the conductor. This calls for an analysis of potentials, electric field strengths, and energies of the shielded point multipoles. This analysis is given below.

Approximation of the electric field of conductors by screened point charges and multipoles

“…If the error of capacitance estimates is such that the error of electric field approximation by fields of point screened charges is acceptable, the approximating charges can be found as (2) 33 2 23 3 0 0 2 22 33 23 …”

“…The last row in the table presents С values numerically calculated in [4]. As can be seen from the data in Table 1, the error in calculating the wire capacitance relative to the screen does not exceed 0.05% when the distance from the wire axis to the screen 2 …”

“…The complex electric potentials and energy relations for the screened point charges and screened point multipoles of arbitrary orders were found in [2]. Kazantsev et al [2] demonstrated that both the complex electric potentials of screened point multipoles and the energy of their interaction could be expressed through derivatives of the complex Green's function. Examples of application of the mathematical screened multipole apparatus for a variational solution of the problem on screened conductor capacitance are considered in the present paper.…”