Does a Curved Mirror Honestly Reflect Your Identity? A Study of Multipole Images in Front of a Grounded Sphere

TL;DR

This paper investigates how multipole charges, such as dipoles and quadrupoles, form images near a grounded sphere, revealing that their images include mixed multipole types and lower-order contributions, expanding electrostatic theory.

Contribution

It uncovers that multipole images near a grounded sphere are not purely of the same type but include lower-order multipoles, providing new insights into electrostatic boundary interactions.

Findings

Image of a multipole includes lower-order multipole contributions.

Multipole images are not necessarily of the same type as the original.

The study extends understanding of electrostatic images beyond flat conductors.

Abstract

The method of image charges is a powerful and elegant technique in electrostatics, commonly used to determine the electric field generated by point charges near conductors of various shapes. While standard problems focus on single charges interacting with conductors, the behavior of multipoles in such configurations has received comparatively less attention, particularly beyond the well-studied case of a flat plane. In this paper, we explore the image formation of electric dipoles and quadrupoles near a conducting sphere and uncover a wonderful result: the image of a given multipole is not necessarily of the same type. Instead, alongside the expected multipole image, the resulting image configuration also includes lower-order multipole contributions. This finding broadens the understanding of electrostatic images and offers new insights into the interaction of multipoles with conducting boundaries.