Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle

Abstract

The quantum electromagnetic zero-point energy of a conducting spherical shell of radius $r$ has been computed to be $\ensuremath{\Delta}E(r)\ensuremath{\cong}\frac{0.09\ensuremath{\hbar}c}{2r}$. The physical reasoning is analogous to that used by Casimir to obtain the force between two uncharged conducting parallel plates, a force confirmed experimentally by Sparnaay and van Silfhout. However, while parallel plates are attracted together because of the zero-point energy, a conducting sphere tends to be expanded. Thus although relevant for the understanding of the quantum-mechanical zero-point energy, the result invalidates Casimir's intriguing model for a charged particle as a charged conducting shell with Poincar\'e stresses provided by the zero-point energy and a unique ratio for $\frac{{e}^{2}}{\ensuremath{\hbar}c}$ independent of the radius.