Lateral Casimir Force between Two Sinusoidally Corrugated Eccentric Cylinders Using Proximity Force Approximation

This paper is devoted to the presentation of the lateral Casimir force between two sinusoidally corrugated eccentric cylinders. Despite that ap-plying scattering matrix method explains the problem exactly, procedure of applying this method is somehow complicated specially at nonzero tem-perature. Using the proximity force approximation (PFA) helps to achieve the lateral Casimir force in a truly explicit manner. We assume the cylin-ders to be slightly eccentric with similar radiuses and separations much smaller than corrugations ’ wave length for the validity of PFA. For such short distances the effect of finite conductivity would be non negligible. In addition to the effect of finite conductivity, we investigate thermal cor-rections of the lateral Casimir force to reduce the inaccuracy of the result obtained by PFA. Assuming the Casimir force density between two par-allel plates, the normal Casimir force between two cylinders is obtained. With the aid of additive summation of the Casimir energy between cylin-ders without corrugation, we obtain the lateral Casimir force between corrugated cylinders