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.Comment: 21 pages, 4 figures; final version accepted by PR