Cylinder-Plate geometry

H.B.G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948)

"The Influence of Retardation on the London-van der Waals Forces"

For asymptotically large separations H, the attractive force between a sphere (radius R) and a plate is

Also, for asymptotically large H, the attractive force between two spheres of radius R is

What is the force between a cylinder (wire) and a plate?

 Pairwise addition of Casimir-Polder forces suggests that (in the limit R << H ):

 Analogy with parallel plates suggests an energy proportional to area:

 Proximity force approximation (exact in the limit R >> H ) gives:

 We find the following exact results (in the limit R >> H ):

due to long wave-length charge fluctuations along the length of the cylinder.

T. Emig, R.L. Jaffe, M. Kardar, and A. Scardicchio, Phys. Rev. Lett. 96, 080403 (2006).

The data points for TM mode come from a numerical Monte Carlo World-line method, by

H. Gies and K. Klingmüller, Phys. Rev. Lett. 96, 220401 (2006)

"Casimir Effect for Curved Geometries: Proximity-Force-Approximation Validity Limits"  

Unexpected non-monotonicity due to three-body effects:

S.J. Rahi, A. Rodriguez, T. Emig, R.L. Jaffe, S.G. Johnson, M. Kardar (2007)

The data points come from a numerical implementation of the stress tensor method, by

A. Rodriguez, M. Ibanescu, D. Iannuzzi, J. D. Joannopoulos, S. G. Johnson, Phys. Rev. A 76, 032106 (2007)

"Virtual photons in imaginary time: Computing exact Casimir forces via standard numerical electromagnetism techniques"