Comment on Superluminality, Parelectricity, and Earnshaws Theorem in Media with Inverted Populations

VOLUME 75, NUMBER 17 PHYSICAL REVIEW LETTERS 23O CTOBER 1995Comment on “Superluminality, Parelectricity, and However, proofs of Earnshaw’s theorem assume that qmoves in a fixed external potential, unmodified by q itself.Earnshaw’s Theorem in Media with InvertedThis is clearly not the case in the example considered byPopulations”Chiao and Boyce [1]. When q is displaced, polarizationcharges are displaced and the electrostatic potential isIn a recent Letter [1] Chiao and Boyce discussedchanged: q therefore cannot be thought of as free tosome theoretical consequences of population inversion inmove in a prescribed external potential. Instead, assumingdielectric media. In particular, they demonstrated that athat the distances between q and the polarization chargescharged particle is in stable equilibrium at the center ofare sufficiently small that displacements of q translatea hollow sphere constructed of a “parelectric” mediumeffectively instantaneously into changes in the potential,having a dielectric constant e,1 . They note that theq always finds itself surrounded by polarization chargesstable equilibrium appears to violate Earnshaw’s theoremacting to force it back to the (stable) equilibrium point at[2,3], which states that such stability cannot be realizedthe center of the sphere.with electrostatic forces alone, and suggest that thisThe conclusion is that Earnshaw’s theorem is of courseseeming violation is due to the fact that their parelectricstrue, but that, ...