Mode-by-mode summation for the zero point electromagnetic energy of an infinite cylinder

Abstract

Using the mode-by-mode summation technique the zero point energy of the electromagnetic field is calculated for the boundary conditions given on the surface of an infinite solid cylinder. It is assumed that the dielectric and magnetic characteristics of the material which makes up the cylinder $(ε_1, μ_1)$ and of that which makes up the surroundings $(ε_2, μ_2)$ obey the relation $ε_1μ_1= ε_2μ_2$. With this assumption all the divergences cancel. The divergences are regulated by making use of zeta function techniques. Numerical calculations are carried out for a dilute dielectric cylinder and for a perfectly conducting cylindrical shell. The Casimir energy in the first case vanishes, and in the second is in complete agreement with that obtained by DeRaad and Milton who employed a Green's function technique with an ultraviolet regulator.