Casimir Force at a Knife's Edge

Abstract

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.

5 pages, 3 figures, uses RevTeX; v2: expanded conclusions; v3: fixed missing factor in Eq. (3) and incorrect diagram label (no changes to results); v4: fix similar factor in Eq. (16) (again no changes to results)

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• Scattering Approach for Fluctuation--Induced Interactions at Fluid Interfaces
• Local and Global Casimir Energies: Divergences, Renormalization, and the Coupling to Gravity
• Geometry and material effects in Casimir physics - Scattering theory
• Something Can Come of Nothing: Surface Approaches to Quantum Fluctuations and the Casimir Force
• Electromagnetic Casimir Forces in Elliptic Cylinder Geometries
• An Investigation of the Casimir Energy for a Fermion Coupled to the Sine-Gordon Soliton with Parity Decomposition
• Semiclassical Estimates of Electromagnetic Casimir Self-Energies of Spherical and Cylindrical Metallic Shells
• Diffractive Effects and General Boundary Conditions in Casimir Energy
• Thermal diffractive corrections to Casimir energies
• Material Dependence of the Wire-Particle Casimir Interaction
• Implications of the Babinet Principle for Casimir Interactions
• Exact Electromagnetic Casimir Energy of a Disk Opposite a Plane
• On the Casimir effect from the zero-point energy: A tangential force and its properties
• Casimir effect: Edges and diffraction
• On the Shape Dependence of the Tangential Casimir Force