Electromagnetic Casimir energy of a disk opposite a plane

Building on work by J. Meixner [Z. Naturforschung 3a, 506 (1948)], we show how to compute the exact scattering amplitude (or $\mathrm{T}$-matrix) for electromagnetic scattering from a perfectly conducting disk. This calculation is a rare example of a nondiagonal $\mathrm{T}$-matrix that can nonetheless be obtained in a semianalytic form. We then use this result to compute the electromagnetic Casimir interaction energy for a disk opposite a plane, for arbitrary orientation angle of the disk, for separations greater than the disk radius. We find that the proximity force approximation (PFA) significantly overestimates the Casimir energy, in the case of both the ordinary PFA, which applies when the disk is parallel to the plane, and the “edge PFA”, which applies when the disk is perpendicular to the plane.

Figure 1

Geometry of the disk for separation $d$, radius $R$, and orientation angle $\theta$.

Figure 2

Scaled Casimir energy for a perfectly conducting disk of radius $R$ opposite a perfectly conducting plane, where the center of the disk is at a distance $d$ from the plane and the normal to the disk is at an angle $\theta$ relative to the normal to the plane. The energies have been scaled by $d^3$ to facilitate comparison.

Figure 3

Casimir energy for a perfectly conducting disk of radius $R$ parallel to a perfectly conducting plane, where the center of the disk is at a distance $d$ from the plane. Black circles represent our results; crosses represent the fluctuating-surface-current calculation from Ref. [32]; the dotted line represents the dipole approximation, given in Eq. (103); and the solid line represents the PFA, given in Eq. (104).

Figure 4

Casimir energy for a perfectly conducting disk of radius $R$ perpendicular to a perfectly conducting plane, where the center of the disk is at a distance $d$ from the plane. Black circles represent our results and the solid line represents the “edge PFA” from Ref. [25].