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 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.PACS numbers: 42.25. Fx, 03.70.+k, Casimir's computation of the force between two parallel metallic plates [1] originally inspired much theoretical interest as a macroscopic manifestation of quantum fluctuations of the electromagnetic field in vacuum. Following its experimental confirmation in the past decade [2], however, it is now an important force to reckon with in the design of microelectromechanical systems [3]. Potential practical applications have motivated the development of numerical methods to compute Casimir forces for objects of any shape [4]. The simplest and most commonly used methods for dealing with complex shapes rely on pairwise summations, as in the proximity force approximation (PFA), which limits their applicability.Recently we have developed a formalism [5,6] that relates the Casimir interaction among several objects to the scattering of the electromagnetic field from the objects individually. (For additional perspectives on the scattering formalism, see references in [6].) This approach simplifies the problem, since scattering is a well-developed subject. In particular, the availability of scattering formulae for simple objects, such as spheres and cylinders, has enabled us to compute the Casimir force between two spheres [5], a sphere and a plate [7], multiple cylinders [8], etc. In this work we show that parabolic cylinders provide another example where the scattering amplitudes can be computed exactly. We then use the exact results for scattering from perfect mirrors to compute the Casimir force between a parabolic cylinder and a plate. In the limiting case when the radius of curvature at its tip vanishes, the parabolic cylinder becomes a semi-infinite plate (a knife's edge), and we can consider how the energy depends on the boundary condition it imposes and the angle it makes to the plane.The surface of a parabolic cylinder in Cartesian coordinates is described by y = (x 2 − R 2 )/2R for all z, as shown in Fig. 1, where R is the radius of curvature at the tip. In parabolic cylinder coordinates [9], defined through x = µλ, y = (λ 2 − µ 2 )/2, z = z, the surface is simply µ = µ 0 = √ R for −∞ < λ, z < ∞. One advantage of the latter coordinate system is that the Helmholtz equationwhich we consider for imaginary wavenumber k = iκ, admits separable solutions. Since sending λ → −λ and µ → −µ returns us to the same point, we restr...

Material dependence of the wire-particle Casimir interaction

“…[25]. In the limit λ p /R c √ ln(2d/R c ) and d d 10 , we reproduce the perfect metal wire-atom interaction energy given in Eq. (34).…”

“…In the retarded limit, d d 10 , we find the Casimir energy between a perfect metal wire and an atom as…”

“…For the plasma wire and an atom in the limit ξ 1 or equivalently λ p /R c √ ln(2d/R c ) and d d 10 , the asymptotic energy reads…”

“…The scattering formalism provides a powerful tool to calculate the Casimir interaction between objects of general shape and material properties [7,8]. There is much recent research activity based on the scattering formalism, e.g., for edges and tips [9,10], anisotropic particles [11], wires and plates [12][13][14][15], spheres and plates [16], and periodic structures [17][18][19]. For some further recent examples, see Ref.…”

An investigation of the Casimir energy for a fermion coupled to the sine-Gordon soliton with parity decomposition

“…The Casimir energy and the resulting forces have been investigated for different fields in different geometries and boundary conditions [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In some of these investigations the Casimir forces on the boundaries are also calculated.…”