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

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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, ...

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VOLUME75, NUMBER17
P H Y S I C A L R E V I E W L E T T E R S
Comment on “Superluminality, Parelectricity, and Earnshaw's Theorem in Media with Inverted Populations”
In a recent Letter [1] Chiao and Boyce discussed some theoretical consequences of population inversion in dielectric media. In particular, they demonstrated that a charged particle is in stable equilibrium at the center of a hollow sphere constructed of a “parelectric” medium having a dielectric constante ,1 note that the. They stable equilibrium appears to violate Earnshaw's theorem [2,3], which states that such stability cannot be realized with electrostatic forces alone, and suggest that this seeming violation is due to the fact that their parelectrics are not in thermal equilibrium. In this Comment we show that Chiao and Boyce's results [1] do not in any way contradict Earnshaw's theorem. Earnshaw's theorem follows from Gauss's law: If we take an isolated pointrin vacuum and draw a Gaussian surfacearoundit,theuxofEthrough this surface must vanish. ThereforeEcannot point everywhere toward or away fromrwould be required for the electrostatic, which stability of a positive or negative charge, respectively, placed atr. LetusbrieyconsideroneoftheexamplesofChiao and Boyce. A point chargeqis at a pointr0inside a spherical vacuum in an otherwise uniform parelectric medium. At any pointroutside the sphere the potential satis es Laplace's equation and can therefore be written [3,4]foutsrdPl`0Clr2sl11dPlscosud, whereuis the angle betweenrandr0measured from the center of the sphere. The potential inside the sphere can likewise be writtenfinsrdqyjr2r0j1Pl`0AlrlPlscosudPl`0fAlrl1qrl,yrs.l11dgPlscosud, wherer,sr.dis the smaller (larger) ofrjrjandr0jr0j the. Using boundary conditions thatfandDr2e ≠fyrare continuous at the surfaceraof the sphere, we easily obtain the following expression for the force acting on the chargeqat positionr0: qE2qf=fsrdgr5r0 2ˆ`lsl11dr02l21 s12 edXl11 2q2qr2012 el0l11∙ ∙ ∙e.sl11da2(1) 2 a3112er0 This result shows that, fore ,1, a chargeqwill be in stable equilibrium at the center of the spherical cavity, in seeming violation of Earnshaw's theorem.
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23 OCTOBER1995
However, proofs of Earnshaw's theorem assume thatq moves in a xedexternal potential, unmodi ed byqitself. This is clearly not the case in the example considered by Chiao and Boyce [1]. Whenqis displaced, polarization charges are displaced and the electrostatic potential is changed:qtherefore cannot be thought of as free to move in a prescribed external potential. Instead, assuming that the distances betweenqand the polarization charges are suf ciently small that displacements ofqtranslate effectively instantaneously into changes in the potential, qalways nds itself surrounded by polarization charges acting to force it back to the (stable) equilibrium point at the center of the sphere. The conclusion is that Earnshaw's theorem is of course true, but that, as Chiao and Boyce have shown, one can nevertheless realize electrostatic stability when the potential in which the charge moves is established by the charge itself. This is not the sort of stability forbidden by Earnshaw's theorem. D. F. V. James and P. W. Milonni Theoretical Division (T-4) Los Alamos National Laboratory Los Alamos,New Mexico 87545 H. Fearn Department of Physics California State University Fullerton,California 92634 Received 10 April 1995 PACS numbers: 41.20.Cv, 41.20.Jb, 42.25.Bs, 42.52.+x
[1] R. Y. Chiao and J. Boyce, Phys. Rev. Lett.73, 3383 (1994). [2] S. Earnshaw, Trans. Cambridge Phil. Soc.7, 97 (1842). [3] J. A. Stratton,Electromagnetic Theory,-warlliHcG(M New York, 1941), p. 116ff. R. Y. Chaio (private commu-nication) has pointed out that Stratton's proof of Earn-shaw's theorem is not entirely convincing in that it allows for a spatially varying dielectric constant and, therefore, an electrostatic potential that does not satisfy Laplace's equation. [4] J. D. Jackson,Classical Electrodynamics(Wiley, New York, 1975), 2nd ed.
Ž 1995The American Physical Society