The aim of this paper is 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, and afterwards, we obtain the effect of deformation on the Casimir interaction energy by considering the sphere to have different kinds of sinusoidal corrugations. We suppose a/r B 0.00755 for the sake of proximity force approximation validity and an experimental accuracy goal of 1 % in the case of a sphere with the radius r at a minimal distance a from a plate. The effect of finite conductivity is taken into account for such short distances. Furthermore, we consider that mirrors are not perfect reflectors indeed which concludes in exact theoretical prediction. Thus, we consider the correction due to finite conductivity. We note that for open geometries, temperature dependence of the Casimir force is larger than that of closed geometries. We also investigate thermal corrections of the Casimir force.
