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We derive analytically the asymptotic behavior of the Casimir interactionbetween a sphere and a plate when the distance between them, $d$, is muchsmaller than the radius of the sphere, $R$. The leading order andnext-to-leading order terms are derived from the exact formula for the Casimirinteraction energy. They are found to depend nontrivially on the dielectricfunctions of the objects. As expected, the leading order term coincides withthat derived using the proximity force approximation. The result on thenext-to-leading order term complements that found by Bimonte, Emig and Kardar[Appl. Phys. Lett. \textbf{100}, 074110 (2012)] using derivative expansion.Numerical results are presented when the dielectric functions are given by theplasma model or the Drude model, with the plasma frequency (for plasma andDrude models) and relaxation frequency (for Drude model) given respectively by9eV and 0.035eV, the conventional values used for gold metal. It is found thatif plasma model is used instead of Drude model, the error in the sum of thefirst two leading terms is at most 2%, while the error in $\theta_1$, the ratioof the next-to-leading order term divided by $d/R$ to the leading order term,can go up to 4.5%.

Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation