The conducting ring viewed as a wormhole

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 that encourage students to think more broadly about the subject. We present our discussion at a level suitable for use in advanced undergraduate and introductory graduate courses on electrostatics, or for use as a supplement to introduce the methods of Riemannian geometry in the context of general relativity.