Casimir force on a solid ball when ε(ω)μ(ω)=1. Finite temperatures

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Abstract
The Casimir surface force on a solid ball is calculated, assuming the material to be dispersive and to be satisfying the condition ε(ω)μ(ω)=1, ε(ω) being the spectral permittivity and μ(ω) the spectral permeability. A recent paper by Brevik and Einevoll [Phys. Rev. D 37, 2977 (1988)] discussed this problem at zero temperature. In this paper, the case of finite temperatures is covered. The analysis is based upon use of the Debye expansion of the modified Bessel functions. This expansion, being an asymptotic one, is limited in accuracy. At general temperatures, the expansion shows an oscillatory variation on the second-order level. Such a variation is unphysical, so that the physical information that one can extract from second-order theory is limited. On the zeroth-order level, the formalism is, however, found to work well. We discuss the zeroth-order approximation in complete form and comment upon the second-order correction. Although numerical methods are in general necessary, useful analytic approximations can be obtained in the limiting cases of very low, or very high, nondimensional temperatures. At low temperatures, the dispersive effect induces a strong, attractive contribution to the force. At high temperatures, the force becomes small.