The Conducting Ring Viewed as a Wormhole

TL;DR

This paper presents a novel geometric approach to compute the Green function for a grounded circular ring in 2D electrostatics by interpreting it as a wormhole, offering new physical insights compared to traditional methods.

Contribution

It introduces a wormhole-based geometric method for calculating Green functions, providing a different physical perspective on image charges in electrostatics.

Findings

Equivalent Green functions from different viewpoints

Distinct physical interpretations of image charges

Connection to earlier work by Sommerfeld and others

Abstract

We compute the exterior Green function for a grounded equi-potential circular ring in two-dimensional electrostatics by treating the system geometrically as a "squashed wormhole" with an image charge located in a novel but obvious position, thereby implementing a method first suggested in 1897 by Sommerfeld. We compare and contrast the strength and location of the image charge in the wormhole picture with that of the conventional point of view where an image charge is located inside the circular ring. While the two viewpoints give mathematically equivalent Green functions, we believe they provide strikingly different physics perspectives. We also comment on earlier Green function results by Hobson in 1900, and by Davis and Reitz in 1971, who applied Sommerfeld's method to analyze a grounded conducting circular disk in three-dimensional electrostatics.