A conducting ball in an axial electric field

TL;DR

This paper analyzes the charge distribution, electric moments, and forces on a conducting ball in an axial electric field, providing theoretical results applicable to inhomogeneous harmonic fields.

Contribution

It offers a comprehensive theoretical framework for calculating charge distribution and electric moments of a conducting sphere in arbitrary harmonic fields, with proofs based on Legendre polynomial properties.

Findings

Derived formulas for charge distribution and electric moments.

Determined the force acting on the conducting ball.

Provided theorems with proofs on the properties of moments.

Abstract

We describe the distribution of a charge, the electric moments of arbitrary order and the force acting on a conducting ball on the axis of the axial electric field. We determine the full charge and the dipole moments of the first order for a conducting ball in an arbitrary inhomogeneous harmonic electric field. All statements are formulated in the form of theorems with proofs basing on properties of the matrix of moments of the Legendre polynomials. The analysis and proof of these properties are presented in Appendix.