Casimir force at a knife's edge

The Casimir force has been computed exactly for only a few simple geometries, such as infinite plates, cylinders, and spheres. We show that a parabolic cylinder, for which analytic solutions to the Helmholtz equation are available, is another case where such a calculation is possible. We compute the interaction energy of a parabolic cylinder and an infinite plate (both perfect mirrors), as a function of their separation and inclination, H and θ, and the cylinder's parabolic radius R. As H/R→0, the proximity force approximation becomes exact. The opposite limit of R/H→0 corresponds to a semi-infinite plate, where the effects of edge and inclination can be probed.