Earnshaw's Theorem

red ball Can a charged body be held in stable stationary equilibrium by electrostatic forces from other charged bodies?

yellow ball "On the Nature of the Molecular Forces which Regulate the Constitution of the Luminiferous Ether,"

Samuel Earnshaw, Trans. Camb. Phil. Soc. 7 97 (1842)

The Castle

The Castle of the Pyrenees: Rene Magritte 1959


red ball  Extension of Earnshaw's theorem to containers with classical mobile charges at finite temperature:

  Now consider the Casimir forces between a set of objects described by position and frequency dependent dielectric response:


yellow ball "Constraints on Stable Equilibria with Fluctuation-Induced (Casimir) Forces,"

S.J. Rahi, M. Kardar, & T. Emig, Phys. Rev. Lett. 105, 070404 (2010)

yellow ballThe Casimir force from a combination of dielectrics cannot lead to stable levitation in equilibrium (zero or finite temperature).

yellow ball Appears to rule out repulsion from metamaterials constructed out of ordinary metals and dielectrics.

yellow ball Levitation can be achieved if conditions for the theorem are violated (cf classical Earnshaw thm.):


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