Casimir interaction between a sphere and a cylinder

TL;DR

This paper derives an exact formula for the Casimir interaction between a sphere and a cylinder under various boundary conditions, revealing that the force is always attractive and providing explicit large separation energy expressions.

Contribution

It generalizes the operator approach to compute the Casimir interaction between a sphere and a cylinder, including explicit translation matrices and energy formulas for different boundary conditions.

Findings

Casimir force is attractive at all separations.

Explicit large separation energy formulas are derived.

Interaction energy scales depend on boundary conditions and geometrical parameters.

Abstract

We study the Casimir interaction between a sphere and a cylinder both subjected to Dirichlet, Neumann or perfectly conducting boundary conditions. Generalizing the operator approach developed by Wittman [IEEE Trans. Antennas Propag. 36, 1078 (1988)], we compute the scalar and vector translation matrices between a sphere and a cylinder, and thus write down explicitly the exact TGTG formula for the Casimir interaction energy. In the scalar case, the formula shows manifestly that the Casimir interaction force is attractive at all separations. Large separation leading term of the Casimir interaction energy is computed directly from the exact formula. It is of order $\sim \hbar c R_1/[L^2\ln(L/R_2)]$, $\sim \hbar c R_1^3R_2^2/L^6$ and $\sim \hbar c R_1^3/[L^4\ln(L/R_2)]$ respectively for Dirichlet, Neumann and perfectly conducting boundary conditions, where $R_1$ and $R_2$ are respectively the radii of the sphere and the cylinder, and $L$ is the distance between their centers.