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The Stable Levitation of Magnets Is Forbid scientific correspondence Magnet levitation at your fingertips he stable levitation of magnets is forbid- Tden by Earnshaw’s theorem, which states that no stationary object made of z stability functions (MB) magnets in a fixed configuration can be r 1Ð1 held in stable equilibrium by any combina- 0 0 tion of static magnetic or gravitational R K v K h forces1–3. Earnshaw’s theorem can be viewed as a consequence of the Maxwell equations, –R which do not allow the magnitude of a C>0 magnetic field in a free space to possess a levitating maximum, as required for stable equilib- Bi cylinder magnet rium. Diamagnets (which respond to mag- –2R netic fields with mild repulsion) are known to flout the theorem, as their negative sus- Figure 2 Levitation at your fingertips. A strong NdFeB ceptibility results in the requirement of a magnet (1.4 tesla) levitates 2.5 metres below a pow- minimum rather than a maximum in the erful superconducting magnet. The field at the levi- 2–4 field’s magnitude . Nevertheless, levitation tation point is about 500 Gauss. of a magnet without using superconductors is widely thought to be impossible. We find where r is the density of the magnet’s that the stable levitation of a magnet can be material and ALJ1.92. Calculation shows achieved using the feeble diamagnetism of that a magnet several millimetres in size materials that are normally perceived as with remnant magnetization of about 1 being non-magnetic, so that even human tesla (NdFeB) can be levitated with a clear- fingers can keep a magnet hovering in mid- Figure 1 A NdFeB magnet (an alloy of neodymium, ance gap, D1d, of several millimetres air without touching it. iron and boron; 4 mm high and 4 mm in diameter) using a 10-cm solenoid and strongly dia- Stable levitation has been demonstrated levitating at the axis of a vertical solenoid of radius magnetic Bi or graphite, in agreement with for diamagnetic objects such as supercon- RLJ10 cm and lengthLJ2R in a magnetic field of 100 our experiment. 2,5–7 ducting pellets and live creatures . Strong gauss. The levitation is stabilized by a bismuth cylin- Equation (2) depends on the product of diamagnetism of superconductors allows der (x411. 5 21014) with inner diameter DLJ8 mm. x and L, which means that by increasing L the situation to be reversed, so that a magnet The photograph shows the top view of the levitating (scaling up the magnet’s size) we can 8 can be levitated above a superconductor . magnet. The right-hand plot shows the stability func- achieve the same D as above using ordinary Paramagnetic objects can also be levitated if tions Kv and Kh calculated for a solenoid with a materials (such as plastic or wood, with 15 placed in a stronger paramagnetic medium, height of twice its radius (solid curves). Diamagnetic xLJǁ10 ). To illustrate this point, we such as ferrofluid or oxygen, which makes interaction C shifts the horizontal stability function Kh show another example of a levitating mag- 9 them effectively diamagnetic . to the left (dashed curve) and a small region of posi- net in Fig. 2 in which human fingers D 4 xLJǁ 15 We set out to lift a magnet by applying a tive U emerges above the point where Kv 0. ( 10 ) are used as diamagnetic stabi- magnetic field and then stabilizing the lizers. Here we use an alternative geometry11 intrinsically unstable equilibrium with The presence of a diamagnetic cylinder in which L is easier to scale up because it is repulsive forces from a nearby diamagnetic results in the last term in equation (1) and, determined not only by the magnet size, but material. We found that, surprisingly, the for the geometry of Fig. 1, we find that also by its strength. The levitating magnet is 4 m x 2 5 m forces created by almost non-magnetic C 45 0| |M /16D , where 0 is the per- placed below a solenoid in the region where materials (susceptibility x of about 1015) meability of free space. If there is no dia- the equilibrium is stable horizontally 4 ¤ * are sufficient to stabilize levitation over dis- magnet (C 0), the stability can never be (Kh 0) but not vertically (Kv 0) (Fig. 1). tances as large as several millimetres under reached (at no point are Kv and Kh both Vertical stability is achieved by means of Earth gravity conditions, even though they positive; Fig. 1). The diamagnetic interac- two horizontal diamagnetic plates (or by decay rapidly with distance as 1/x5 (Fig. 1). tion allows the energy U to have a mini- the fingertips). ¤ & ¤ For stable levitation, an equilibrium mum (Kv 0 and Kh C 0) which In this geometry, the positive constant 8 ¤ 8 2 4 m ᎂxᎂ 2 p 5 requires that the magnetic force MB (z) emerges for C MB (z) /8B(z) just above C 6 0 M / D counters Kv and the levi- compensates the gravitational force mg, the point of a maximum field gradient tation condition is similar to equation (2), where M is the magnetic moment and B(z) (B88(z)40). It is counterintuitive that levi- except that now D denotes the separation and B8(z) are the magnetic field on the axis tation is easiest in the most inhomogeneous between the plates, ALJ1.02 and L≡4B8/B88 and its derivative, respectively. For the equi- field region, rather than in the centre of a is approximately the distance from the cen- librium to be stable, it must be in a region solenoid where the field is almost uniform. tre of a solenoid to a levitating magnet. The where the total energy of the magnet It is instructive to introduce a character- larger the distance, the easier it is to stabilize 41 & & U MB(r) mgz Udia has a minimum istic scale L on which the field changes: levitation by diamagnetic repulsion. L is D ¤ 84 ( U 0), where Udia is the energy of dia- B B/L. At the optimum levitation point limited by the requirement on the field gra- magnetic interaction with the cylinder. (B88(z)40), L varies between R and 1.2R dient, B8(z)4mg/M. To reach such a large Close to the equilibrium position at the for long and short solenoids, respectively. If L, as in Fig. 2, we used an 11-tesla supercon- field axis3,10, we approximate our levitating magnet by a ducting solenoid a metre in diameter. If LJ & 1 8 & 2& 2& U U0 [mg MB (z)]z Kvz Khr sphere of diameter d with a remnant field stronger diamagnets are used (such as 2& 4 p m 3 Cr … (1) Br, then M ( /4 0)Brd , and the require- graphite or bismuth), this type of levitation ≡1 88 where Kv(z) MB (z)/2 and ment for levitation becomes can also be achieved with small permanent ≡1 8 21 88 ᎂxᎂ 2 3 m r 1/5¤ ¤ Kh(z) M[B (z) 2B(z)B (z)]/8B(z). A( LBr d / 0 g) D d (2) magnets, making miniature hand-held NATURE | VOL 400 | 22 JULY 1999 | www.nature.com 323 scientific correspondence devices accessible to everyone (M. D. S., unpublished data). These could replace the a mN mN b mN c existing servo levitation devices for some applications. A. K. Geim*, M. D. Simon†, M. I. Boamfa*, mN" mN" L. O. Heflinger† mN" *High Field Magnet Laboratory, University of Nijmegen, 6525 ED Nijmegen, The Netherlands mShore mShore mShore e-mail: [email protected] d e f †Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA 1. Earnshaw, S. Trans. Camb. Phil. Soc. 7, 97–112 (1842). 2. Brandt, E. H. Science 243, 349–355 (1989). P< 0.001 P< 0.001 3. Berry, M. V. &. Geim, A. K. Eur. J. Phys. 18, 307–313 (1997). 4. Thomson, W. (Lord Kelvin) Reprints of Papers on Electrostatics mShore mShore and Magnetism (Macmillan, London, 1872). g h 5. Braunbeck, W. Z. Phys. 112, 753–763 (1939). 6. Beaugnon, E. & Tournier, R. Nature 349, 470 (1991). 7. Geim, A. Phys. Today 51, 36–39 (September 1998). 8. Arkadiev, A. Nature 160, 330 (1947). 9. Ikezoe, Y. et al. Nature 393, 749–750 (1998). 10.Simon, M. D. et al. Am. J. Phys. 65, 286–292 (1997). P< 0.001 11.Boerdijk, A. H. Philips Tech. Rev. 18, 125–127 (1956). Figure 1 Effects of long-wavelength light and head caps on bimodal magnetic orientation in newts. a–c, Pre- dicted orientation of newts (double-headed arrow) and their perception of the direction of the magnetic field (single-headed arrow)5,6. Training tanks have the shore towards magnetic north (mN); circular test arenas Extraocular magnetic show the predicted response of the newts under either full-spectrum (beige) or long-wavelength light (yel- low). a, Full-spectrum training and testing: newts should perceive the shore to be towards magnetic north compass in newts and exhibit bimodal magnetic orientation along the shoreward axis. b, Full-spectrum training, long-wavelength testing: newts’ perception of magnetic north in testing, and their orientation in the test arena, should be rotated Geomagnetic orientation is widespread 90º (mN8) from magnetic north during training. c, Long-wavelength training and testing: newts’ perception of among organisms, but the mechanism(s) of the magnetic field should be rotated 90º relative to the actual field during training and testing. Their percep- magnetoreception has not been identified tion of the magnetic field in the arena would be the same as in the outdoor tank.
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