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Magnetic levitation induced by negative permeability A. A. Rangelov Department of Physics, Sofia University, James Bourchier 5 Blvd, 1164 Sofia, Bulgaria Inthispaperwestudytheinteraction betweenapointmagneticdipoleandasemi-infinitemeta- material using the method of images. We obtain analytical expressions for the levitation force for an arbitrarily oriented dipole. Surprisingly the maximal levitation force for negative permeability is found to be stronger compared to thecase when the dipole is above a superconductor. PACSnumbers: 41.20.Gz, 81.05.Zx,02.70.Pt,73.20.Mf 1 1 0 Naturallymaterialshavepositivepermeability,butfor J 2 the last decade artificial materials with negative perme- m source ability(metamaterials)attractgrowinginterestfromthe- n m a oreticalandexperimentalperspectives[1,2]. Insuchma- medium 1 J terials the permeability µ/µ0 (µ0-vacuum permeability) d 4 is given by ] µ/µ0 =1−αω02/ ω2−ω02 , (1) r he whereω0 isanalogtotheplasma(cid:0)frequenc(cid:1)y(called“mag- mediumm2 d neticplasmafrequency”)andαistheso-calledfillingfac- t o tor[1,2]. Itisnothardtocheckthatforsomefrequencies m' image . t ω the permeability µ is negative. J a Most of the papers, related to negative permeability, m focusontheopticalpropertiesofthosemetamaterials[3– - 5]. In the present paper we consider the magnetic forces d n in the presence of material with negative permeability. FIG. 1: Magnetic dipole m embedded in a semi-infinite ma- o Weexploretheoreticallytheforceactingonapointmag- terialwithpermeabilityµ1 atadistancedawayfromaplane c netic dipole placed next to a materialwith negative per- interface that separates the two mediums and its dipole im- [ meability. Our examination relies on the Pendry’s cri- age m′ symmetrically placed in a semi-infinite material with 1 terion for validity of magnetostatic limit in the present permeability µ2. v problem [6]: the wavelength, corresponding to the fre- 8 quency ω in Eq. (1), should be longer than the distance 3 Using the well-known expression for the potential en- between the magnetic dipole and the material with neg- ′ 8 ergyoftworealmagneticdipolesmandm atadistance ativepermeability. Usingthemethodofimages[7–9],we 0 r [10, 11] show that for negative permeability materials the force . 01 between the metamaterial and the magnetic dipole can 3µ1 m.m′ 3(r·m)(r.m′) be attractiveor repulsive andcanevenhavea muchbig- V = − , (3) 1 ger value compared to the conventional materials. 4π " |r|3 |r|5 # 1 We consider a point magnetic dipole m in media with : we obtain the following expression for the vertical force v permeability µ1, oriented at angle θ with respect to the i acting on a dipole m X axisperpendiculartotheplanesurfaceofamediumwith r permeability µ2 as shown in Fig. 1. The distance from 3µ1 1+cos2θ mm′ 3µ1m2 1+cos2θ µ1−µ2 a tishedd(ispeoeleFtigo.th1)e.suTrhfaecper,otbhlaetmsewpaersaetteitshteotwfinodmheodwiumthse, F = 4π (cid:0) (2d)4(cid:1) = 4π (cid:0) (2d)4 (cid:1)µ1+µ2, (4) dipole m interacts with the medium µ2. The solution which force is perpendicular to the surface of a medium is quite straightforward when one uses the method of with permeability µ2 and is attractive or repulsive de- images[7–9], whichstatesthatthe magnetic potentialin pendingontheratio µ21−µ1µ2 /(µ1+µ2). Therepul- the µ1 region is a superposition of the dipole fields from sion force for conventional materials is strongest when ′ m and the image m (cid:0) (cid:1) the magnetic dipole m is in free space, pointing towards ′ µ1−µ2 thesurfaceplaneandisplacednexttoasuperconductor. m = m, (2) µ1+µ2 Thisis the Meissner’seffect[9,12–14],whichfollowsfor- mallyfromEq. (4)by setting θ =0, µ1 =µ0 andµ2 =0 placed at the same distance d on the other side of the in this case the vertical lifting force is plane separating the two mediums and oriented at the same angle θ with the perpendicular axis to the plane 3µ0m2 f = . (5) surface (see Fig. 1). 2π(2d)4 2 5 5 Eq. (6) has singularity for µ = −µ0. One may think 4 4 that it leads to an infinite force, but one should note thatdispersionlessmaterialisanabstractionandafinite 3 3 positiveimaginarycomponentofthepermeabilityalways 2 2 exists[6]. The imaginarycomponentofpermeabilitycan s f) 1 1 no longer be neglected when the difference between the unit 0 0 real permeabilities is small, which resolves the infinite n force paradox. F (i -1 -1 -2 -2 Even more interesting than the direction of the force -3 -3 is that the force could be stronger than the one due to the Meissner effect, which can be seen from Fig. 2. For -4 -4 example, when µ = −2µ0 the dipole is attracted to the -5 -5 metamaterial with maximal force -10 -8 -6 -4 -2 0 2 4 6 8 10 m (in units m ) 0 9µ0m2 F =− =−3f. (9) FIG. 2: Maximal vertical force, the case θ =0 from Eq. (6), 4 2π(2d) as a function of the permeability µ. The force is divided by the maximal force f, which is due to Meissner’s effect from Eq. (5). The lastequationand Fig. 2 makes it clear that the levitation force acting on a magnetic dipole can be much Ifwe considerthe casewhen the firstmediais vacuum stronger than the levitation force on the same dipole (µ1 = µ0) while the second media is material with per- above a superconductor. Moreoverthe levitation in case meability µ2 = µ (material or metamaterial), the force of metamaterials can be done at room temperature, in acting on a magnetic dipole is contrast to the superconductor levitation case. 3µ0m2 1+cos2θ µ0−µ The examined force of a point magnetic dipole above F = . (6) 4π (cid:0) (2d)4 (cid:1)µ0+µ a negative permeability has the potential to be not only a curiousandintriguing example ofwhatartificialmate- This force is repulsive when rials can do, but also a useful and effective technique to levitate small objects and therefore to make frictionless |µ|<µ0 (7) devices. and attractive when This work has been supported by the European Com- missionprojects EMALI and FASTQUAST, the Bulgar- |µ|>µ0. (8) ian NSF grants D002-90/08and DMU02-19/09. [1] J.B.Pendry,A.J.Holden,D.J.Robbins,W.J.Stewart, [8] J. D.Jackson, Classical Electrodynamics (JphnWiley & IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). Sons, New York,1998). [2] M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. [9] H.E.Knoepfel,MagneticFields: AComprehensive The- Larkman, D. J. Gilderdale, J. V. Hajnal, Science 291, oretical Treatise for Practical Use (Jphn Wiley & Sons, 849 (2001). New York,2000). [3] J. B. Pendry,Phys. Rev.Lett. 85, 3966 (2000). [10] Z.J.Yang,T.H.Johansen,G.Helgesen,A.T.Skjeltorp, [4] M.W.Klein,C.Enkrich,M.Wegener,S.Linden,Science Physica C 160, 461 (1989). 313, 502 (2006). [11] M. W. Coffey,Phys. Rev.B 65, 214524 (2002). [5] W. H. Wee and J. B. Pendry, New J. Phys. 11, 073033 [12] A. A.Kordyuk,J. Appl. Phys.83, 610 (1998). (2009). [13] Qiong-Gui Lin, Phys. Rev.B. 74, 024510 (2006). [6] J. B. Pendry,Contemp. Phys. 45, 191 (2004). [14] J.L.Perez-DiazandJ.C.Garcia-Prada,PhysicaC467, [7] D.J.Griffiths,IntroductiontoElectrodynamics (Prentice 141 (2007). Hall, 1998).