Repulsion/Levitation?

1. Boundary Conditions & Exotic materials

red ball Similar materials (boundary conditions) attracts, but sufficiently dissimilar boundaries may repel.

yellow ball  In Lifshitz (DLP) theory, unlike boundaries are obtained through an intervening medium with dielectric constant intermediate to the sides, as in the case of liquid helium climbing (wetting) a wall.  (dielectric constants: solid > helium > air).

Rollin

yellow ball "Verification of Lifshitz theory of the van der Waals Potential using liquid-helium films,"

E.S. Sabisky and C.H. Anderson, Phys. Rev. A 7, 790 (1973)

yellow ball "Measured long-range repulsive Casimir–Lifshitz forces,"

J. N. Munday, F. Capasso & V. A. Parsegian, Nature 457, 170 (2009) (gold, bromobenzene, silica)

yellow ball  Immersing MEMs in fluids is not practical.  Is repulsion across vacuum possible?

yellow ball"Van der Waals forces and zero-point energy for dielectric and permeable materials,"

T.H. Boyer, Phys. Rev. A 9, 2078 (1974) (large permittivity and large permeability repel)

yellow ballA material with large permeability is required for repulsion, but in ordinary materials permeability is close to one.

yellow ballMetamaterials, incorporating arrays of microengineered circuitry mimic, at certain frequencies, a strong magnetic response and have been proposed as candidates for Casimir repulsion across vacuum.

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2. Repulsion via Geometry & Shape

red ballAbraham-Lorentz (+Casimir) model of electron: A spherical conducting shell of radius R , with charge e

[ "Introductory remarks on quantum electrodynamics," H.B.G. Casimir, Physics 19, 846 (1956) ]

yellow ball Balancing the repulsive Coulomb energy, with the (presumed attractive) Casimir energy:

yellow ball "Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle," T.H. Boyer, Phys. Rev. 174, 1764 (1968)

2A = - 0.09235  is negative!  [also obtained by R. Balian and B. Duplantier (1977)]

yellow ballNot only for spheres, but also rectangular boxes:

["Vacuum means of energy-momentum ...,"  S.G. Mamaev and N.N. Trunov, Sov. Phys. J. (USA) 22, 51 (1979)]

The "Casimir energy" of a parallelepiped changes sign with aspect ratio:

red ball This putative repulsive force has been proposed as yet another means of levitation:

       ,     but

yellow ball "Attractive Casimir forces in a closed geometry,"

M. P. Hertzberg, R. L. Jaffe, M. Kardar, and A. Scardicchio, Phys. Rev. Lett. 95, 250402 (2005):

In the physically accessible geometry of a piston, the partition is always attracted to the closer side.

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yellow ball "Opposites Attract: A Theorem about the Casimir Force,"

O. Kenneth and I. Klich, Phys. Rev. Lett. 97, 160401 (2006) (two halfs of a cut sphere attract)

red ball Constrained repulsion?

yellow ball "Casimir repulsion between metallic objects in vacuum,"

M. Levin, A.P. McCauley, A.W. Rodriguez, M.T.H. Reid, S.G. Johnson,  Phys. Rev. Lett. 105, 090403 (2010)