# Casimir Force at a Knife’s Edge

### Noah Graham 1, Alexander Shpunt 2, Thorsten Emig 2, 3, 4, Sahand Jamal Rahi 2, Robert L. Jaffe 2, 5, Mehran Kardar 2

#### Physical Review D 81 (2010) 061701

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 $\theta$, and the cylinder's parabolic radius $R$. As $H/R\to 0$, the proximity force approximation becomes exact. The opposite limit of $R/H\to 0$ corresponds to a semi-infinite plate, where the effects of edge and inclination can be probed.

• 1. Middlebury College,
Middlebury Colleg
• 2. Department of Physics,
Massachusetts Institute of Technology
• 3. Institut für Theoretische Physik,
Universität zu Köln
• 4. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 5. Center for Theoretical Physics and Laboratory for Nuclear Science,
Massachusetts Institute of Technology