Casimir energy of a compact cylinder under the condition $\epsilon\mu = c^{-2}$

TL;DR

This paper calculates the Casimir energy for a compact cylinder in a medium with matched light velocity, revealing that the energy expansion starts at the fourth power of a specific material contrast parameter.

Contribution

It provides explicit formulas for the Casimir energy of a cylinder under the condition of equal light velocity and analyzes its behavior compared to other geometries.

Findings

Casimir energy expansion begins with ourth power of b4b7

Casimir forces are attractive for the cylinder

Explicit formulas enable numerical calculations for any b4b7

Abstract

The Casimir energy of an infinite compact cylinder placed in a uniform unbounded medium is investigated under the continuity condition for the light velocity when crossing the interface. As a characteristic parameter in the problem the ratio $\xi^2=(\epsilon_1-\epsilon_2)^2/ (\epsilon_1+\epsilon_2)^-2 = (\mu_1-\mu_2)^2/(\mu_1+ \mu_2)^2 \le 1$ is used, where $\epsilon_1$ and $\mu_1$ are, respectively, the permittivity and permeability of the material making up the cylinder and $\epsilon_2$ and $\mu_2$ are those for the surrounding medium. It is shown that the expansion of the Casimir energy in powers of this parameter begins with the term proportional to $\xi^4$. The explicit formulas permitting us to find numerically the Casimir energy for any fixed value of $\xi^2$ are obtained. Unlike a compact ball with the same properties of the materials, the Casimir forces in the problem under consideration are attractive. The implication of the calculated Casimir energy in the flux tube model of confinement is briefly discussed.