Abstract
We calculate the exact Casimir interaction energy between two perfectly conducting, very long, eccentric cylindrical shells using a mode summation technique. Several limiting cases of the exact formula for the Casimir energy corresponding to this configuration are studied both analytically and numerically. These include concentric cylinders, cylinder-plane and eccentric cylinders, for small and large separations between the surfaces. For small separations we recover the proximity approximation, while for large separations we find a weak logarithmic decay of the Casimir interaction energy, typical of cylindrical geometries.
Highlights
We have derived an exact formula for the Casimir interaction energy between eccentric cylinders using a mode summation technique
This formula is written as an integral of the determinant of an infinite dimensional matrix, and it reproduces as a particular case the interaction energy between concentric cylinders, and as a limiting case the energy in the cylinder–plane geometry
We have carried out the numerical evaluation of the Casimir interaction energy using both the exact and tridiagonal formulae, and studied different limiting cases of relevance for Casimir force measurements
Summary
P where wp are the eigenfrequencies of the electromagnetic field satisfying perfect conductor boundary conditions on the surfaces of the conductors, and wp are those corresponding to the reference vacuum (conductors at infinite separation). We use this result to replace the sum over n in Equation (3) by a contour integral. In the rest of this section, we will consider the particular configuration of two eccentric cylinders with circular sections of radii a and b, respectively. It is convenient to compute the difference between the energy of the system of two eccentric cylinders and the energy of two isolated cylindrical shells of radii a and b, E12(σ) = Ec(σ) − E1(σ, a) − E1(σ, b),. F1cyl(a) is a function that vanishes at the eigenfrequencies of an isolated cylindrical shell of radius a. In order to compute Ec(σ), E1(σ, a) and E1(σ, b) separately, an adequate contour is a circular segment C and two straight line segments forming an angle φ and π − φ with respect to the imaginary axis (see figure 2). It is worth noting that the structure of this expression is similar to the ones derived recently for the cylinder–plane geometry using path integrals [11, 12], and for the sphere–plane geometry using the Krein formula [10]
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