We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading-order and next-to-leading-order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading-order term coincides with that derived using the proximity force approximation. Numerical results are presented when the dielectric functions are given by the plasma model or the Drude model, with the plasma frequency (for plasma and Drude models) and relaxation frequency (for Drude model) given by the conventional values used for gold metal. It is found that if plasma model is used instead of the Drude model, the error in the sum of the first two leading terms is at most 2%, while the error in ${\theta }_1$, the ratio of the next-to-leading-order term divided by $d/R$ to the leading-order term, can go up to 4.5%.

• Received 21 March 2013

DOI:https://doi.org/10.1103/PhysRevD.88.045019