Non-contact gears: I. Next-to-leading order contribution to lateral Casimir force between corrugated parallel plates

Abstract

We calculate the lateral Casimir force between corrugated parallel plates, described by $\delta$-function potentials, interacting through a scalar field, using the multiple scattering formalism. The contributions to the Casimir energy due to uncorrugated parallel plates is treated as a background from the outset. We derive the leading- and next-to-leading-order contribution to the lateral Casimir force for the case when the corrugation amplitudes are small in comparison to corrugation wavelengths. We present explicit results in terms of finite integrals for the case of the Dirichlet limit, and exact results for the weak-coupling limit, for the leading- and next-to-leading-orders. The correction due to the next-to-leading contribution is significant. In the weak coupling limit we calculate the lateral Casimir force exactly in terms of a single integral which we evaluate numerically. Exact results for the case of the weak limit allows us to estimate the error in the perturbative results. We show that the error in the lateral Casimir force, in the weak coupling limit, when the next-to-leading order contribution is included is remarkably low when the corrugation amplitudes are small in comparison to corrugation wavelengths. We expect similar conclusions to hold for the Dirichlet case. The analogous calculation for the electromagnetic case should reduce the theoretical error sufficiently for comparison with the experiments.

25 pages, 10 figures, appendix added, references corrected, typos corrected

References in corpus (6)

• Casimir Forces between Compact Objects: I. The Scalar Case
• Lateral Casimir force beyond the Proximity Force Approximation
• Multiple Scattering Methods in Casimir Calculations
• Non-contact gears: II. Casimir torque between concentric corrugated cylinders for the scalar case
• Comment on ``Lateral Casimir force beyond the proximity-force approximation''
• Reply to ``Comment on ``Lateral Casimir Force beyond the Proximity Force Approximation'' ''

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