# Casimir Energies for Isorefractive or Diaphanous Balls

## TL;DR

This paper investigates the Casimir energies of isorefractive or diaphanous spherical objects, exploring their finiteness and divergences, and extends the analysis to electromagnetic delta-function spheres with mixed electric and magnetic couplings.

## Contribution

It re-examines the Casimir energy of isorefractive spheres and explores the extension to delta-function spheres with equal and opposite electric and magnetic couplings.

## Key findings

- Finite self-energy for isorefractive spheres when inside and outside light speeds match
- Divergences appear in delta-function sphere models, complicating energy extraction
- Some limited results obtained despite divergence issues

## Abstract

It is familiar that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with $\varepsilon\mu=1$, a so-called isorefractive or diaphanous ball. Here we re-examine that example, and attempt to extend it to an electromagnetic $\delta$-function sphere, where the electric and magnetic couplings are equal and opposite. Unfortunately, although the energy expression is superficially ultraviolet finite, additional divergences appear that render it difficult to extract a meaningful result in general, but some limited results are presented.