Casimir Energy between a Sinusoidally Corrugated Sphere and a Plate Using Proximity Force Approximation

In this paper aiming to obtain the Casimir energy between a sinusoidally corrugated sphere and a plate, we first present a derivation of the sphere-plate Casimir force obtained by applying proximity force approximation (PFA). One may investigate this problem by using the scattering matrix method. Despite that this method achieves the exact result, but difficulties of applying this formalism make the procedure complicated. We use the PFA to achieve the purpose in a truly explicit manner. Considering the sphere to have different kinds of sinusoidally corrugations, we obtain the effect of deformation on the Casimir interaction energy. We suppose that $ a/r \leqslant 0.00755$ for the validity of PFA and an experimental accuracy goal of $1\verb"%"$ in the case of sphere with radius $r$ at a minimal distance $a$ from a plate \cite{0601094}. The effect of finite conductivity would be non negligible for such short distances. To reduce the inaccuracy of the result obtained by PFA, we have also investigate thermal corrections of the Casimir force.