Abstract. For the configuration of a sphere in front of a plane we calculate the first two terms of the asymptotic expansion for small separation of the Casimir force. We consider both Dirichlet and Neumann boundary conditions.

We recalculate the first analytic correction beyond proximity force approximation for a sphere in front of a plane for a scalar field and for the electromagnetic field. We use the method of Bordag and Nikolaev [J. Phys. A 41, 164002 (2008)]. We confirm their result for Dirichlet boundary conditions whereas we find a different one for Robin, Neumann and conductor boundary conditions. The difference can be traced back to a sign error. As a result, the corrections depend on the Robin parameter. Agreement is found with a very recent method of derivative expansion

We consider the vacuum energy for a configuration of a sphere in front of a plane, both obeying conductor boundary condition, at small separation. For the separation becoming small we derive the first next-to-leading order of the asymptotic expansion in the separation-to-radius ratio ε. This correction is of order ε. In opposite to the scalar cases it contains also contributions proportional to logarithms in first and second order, ε ln ε and ε(ln ε) 2 . We compare this result with the available findings of numerical and experimental approaches.

We consider the vacuum energy for a configuration of two cylinders and obtain its asymptotic expansion if the radius of one of these cylinders becomes large while the radius of the other one and their separation are kept fixed. We calculate explicitly the next-to-leading order correction to the vacuum energy for the radius of the other cylinder becoming large or small.