Casimir potential of a compact object enclosed by a spherical cavity

Saad Zaheer, Sahand Jamal Rahi, Thorsten Emig, and Robert L. Jaffe

Phys. Rev. A 82, 052507 – Published 10 November 2010

Abstract

We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects’ scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to the Casimir-Polder result. Although it is not a simple power law, the dependence of the force on the separation of the object from the containing sphere is a universal function of its displacement from the center of the sphere, independent of other details of the object’s electromagnetic response. Furthermore, we compute the exact Casimir force between two metallic spheres contained one inside the other at arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the leading-order correction to the proximity force approximation for two metallic spheres both outside and within one another.

Received 24 August 2010

DOI:https://doi.org/10.1103/PhysRevA.82.052507

Authors & Affiliations

Saad Zaheer^1,*, Sahand Jamal Rahi¹, Thorsten Emig^2,3, and Robert L. Jaffe⁴

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Institut für Theoretische Physik, Universität zu Köln, Zülpicher Strasse 77, D-50937 Köln, Germany
³Laboratoire de Physique Théorique et Modèles Statistiques, CNRS UMR 8626, Université Paris-Sud, F-91405 Orsay, France
⁴Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*Present address: Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA.

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Vol. 82, Iss. 5 — November 2010

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Images

Figure 1
Interior geometry. An object inside the cavity of an external object. We assume that it is possible to choose a bounding sphere that does not overlap with the surface of the cavity. The distance between the two origins is denoted by $X_{ie}$ .Reuse & Permissions
Figure 2
Casimir energy between two conducting spheres. The black line shows the Casimir energy, $R (x) = E / E_{fPFA}$ , as a function of $x = a / (R - r)$ , where $a$ is the displacement of centers. The radius of the inner sphere is fixed at $r = 0.5 R$ , where $R$ is the radius of the outer sphere. In the limit $x \to 1$ , the Casimir energy approaches the PFA energy, which is marked by the dashed horizontal line. $E_{fPFA}$ denotes the “full” form of the PFA energy discussed in Sec. 4. At intermediate separations, the Casimir energy is dominated by lower partial waves. For example, the gray line shows that the energy obtained by integrating Eq. (2) to partial wave order $l = 25$ is accurate up to $x ~ 0.7$ . The black line is obtained by extrapolating to $l = \infty$ . (Inset) Convergence at close separations, $0.75 ⩽ x ⩽ 1$ .Reuse & Permissions
Figure 3
Casimir force between two conducting spheres. The black line shows the Casimir force, $F / F_{fPFA}$ , between two conducting spheres as a function of $x = a / (R - r)$ where $a$ is the displacement of their centers. The radius of the inner sphere is fixed at $r = 0.5 R$ , where $R$ is the radius of the outer sphere. In the limit $x \to 1$ , the Casimir force approaches the PFA. $F_{fPFA}$ denotes the “full” form of the PFA discussed in Sec. 4.Reuse & Permissions
Figure 4
PFA correction coefficients for spheres. $r / R$ ranges from $- 1$ (interior concentric), to zero (sphere-plane), to $+ 1$ (exterior, equal radii). The data points correspond to the exact values of $θ_{1}$ , calculated numerically, while the solid black curve is a fit (see text). (Inset) “Interior” and “exterior” geometrical configurations.Reuse & Permissions
Figure 5
Plot of the functions $f^{M / E} (a / R)$ and $g^{M / E} (a / R)$ , defined in Eqs. (14) and (15), respectively.Reuse & Permissions
Figure 6
Comparison of the interior Casimir-Polder result with the exact Casimir energy predicted by Eq. (2) for conducting spheres. Plotted along the $y$ axis is the fractional error $(E - E_{CP}) / E$ as a percentage, where $E_{CP}$ is calculated from Eq. (18). For each value of the separation $a / R$ (denoted by point markers listed in the figure), we have plotted the fractional difference between $E$ and $E_{CP}$ at the leading order $O (r^{3} / R^{3})$ (gray) and next-to-leading order $O (r^{5} / R^{5})$ (black) as a function of the internal sphere radius $r / R$ .Reuse & Permissions

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