Non-contact gears: II. Casimir torque between concentric corrugated cylinders for the scalar case

TL;DR

This paper calculates the Casimir torque between concentric corrugated cylinders using a scalar field model, providing analytic and exact results in specific limits, relevant for non-contact gear mechanisms.

Contribution

It derives explicit analytic expressions for Casimir torque between corrugated cylinders, including Dirichlet and weak coupling cases, extending understanding of non-contact gear interactions.

Findings

Analytic expressions for Casimir torque with small corrugation amplitudes.

Exact results in the weak coupling limit.

Torque expressions approach parallel plate results for large cylinder radii.

Abstract

The Casimir interaction between two concentric corrugated cylinders provides the mechanism for non-contact gears. To this end, we calculate the Casimir torque between two such cylinders, described by $\delta$-potentials, which interact through a scalar field. We derive analytic expressions for the Casimir torque for the case when the corrugation amplitudes are small in comparison to the corrugation wavelengths. We derive explicit results for the Dirichlet case, and exact results for the weak coupling limit, in the leading order. The results for the corrugated cylinders approach the corresponding expressions for the case of corrugated parallel plates in the limit of large radii of cylinders (relative to the difference in their radii) while keeping the corrugation wavelength fixed.