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The dielectric sphere has been an important test case for understanding and calculating the vacuum force of a dielectric body onto itself. Here we develop a method for computing this force in homogeneous spheres of arbitrary dielectric properties embedded in arbitrary homogeneous backgrounds, assuming only that both materials are isotropic and dispersionless. Our results agree with known special cases; most notably we reproduce the prediction of Boyer and Schwinger et al. of a repulsive Casimir force of a perfectly reflecting shell. Our results disagree with the literature in the dilute limit. We argue that Casimir forces can not be regarded as due to pair-wise Casimir-Polder interactions, but rather due to reflections of virtual electromagnetic waves.
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It has always been conventionally understood that, in the dilute limit, the Casimir energy of interaction between bodies or the Casimir self-energy of a dielectric body could be identified with the sum of the van der Waals or Casimir-Polder energies of the constituents of the bodies. Recently, this proposition for self-energies has been challenged by Avni and Leonhardt [Ann. Phys. {bf 395}, 326 (2018)], who find that the energy or self-stress of a homogeneous dielectric ball with permittivity $varepsilon$ begins with a term of order $varepsilon-1$. Here we demonstrate that this cannot be correct. The only possible origin of a term linear in $varepsilon-1$ lies in the bulk energy, that energy which would be present if either the material of the body, or of its surroundings, filled all space. Since Avni and Leonhardt correctly subtract the bulk terms, the linear term they find likely arises from their omission of an integral over the transverse stress tensor.
In our paper [Ann. Phys. (NY) 395, 326 (2018)] we calculate the Casimir stress on a sphere immersed in a homogeneous background, assuming dispersionless dielectrics. Our results appear to challenge the conventional picture of Casimir forces. The paper [arXiv:1909.05721] criticises our approach without offering an alternative. In particular, the paper [arXiv:1909.05721] claims that we have made an unjustified mathematical step. This brief comment clarifies the matter.
We derive an exact solution for the Casimir force between two arbitrary periodic dielectric gratings and illustrate our method by applying it to two nanostructured silicon gratings. We also reproduce the Casimir force gradient measured recently [1] between a silicon grating and a gold sphere taking into account the material dependence of the force. We find good agreement between our theoretical results and the measured values both in absolute force values and the ratios between the exact force and PFA predictions.
The Casimir effect in an inhomogeneous dielectric is investigated using Lifshitzs theory of electromagnetic vacuum energy. A permittivity function that depends continuously on one Cartesian coordinate is chosen, bounded on each side by homogeneous dielectrics. The result for the Casimir stress is infinite everywhere inside the inhomogeneous region, a divergence that does not occur for piece-wise homogeneous dielectrics with planar boundaries. A Casimir force per unit volume can be extracted from the infinite stress but it diverges on the boundaries between the inhomogeneous medium and the homogeneous dielectrics. An alternative regularization of the vacuum stress is considered that removes the contribution of the inhomogeneity over small distances, where macroscopic electromagnetism is invalid. The alternative regularization yields a finite Casimir stress inside the inhomogeneous region, but the stress and force per unit volume diverge on the boundaries with the homogeneous dielectrics. The case of inhomogeneous dielectrics with planar boundaries thus falls outside the current understanding of the Casimir effect.
Our previous article [Phys. Rev. Lett. 104, 060401 (2010)] predicted that Casimir forces induced by the material-dispersion properties of certain dielectrics can give rise to stable configurations of objects. This phenomenon was illustrated via a dicluster configuration of non-touching objects consisting of two spheres immersed in a fluid and suspended against gravity above a plate. Here, we examine these predictions from the perspective of a practical experiment and consider the influence of non-additive, three-body, and nonzero-temperature effects on the stability of the two spheres. We conclude that the presence of Brownian motion reduces the set of experimentally realizable silicon/teflon spherical diclusters to those consisting of layered micro-spheres, such as the hollow- core (spherical shells) considered here.
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