M.V. Berry, A.K. Geim. OF FLYING FROGS AND LEVITRONS

Abstract

Diamagnetic objects are repelled by magnetic fields. If the fields are strong enough, this repulsion can balance gravity, and objects levitated in this way can be held in stable equilibrium, apparently violating Earnshaw’s theorem. In fact Earnshaw’s theorem does not apply to induced magnetism, and it is possible for the total energy (gravitational + magnetic) to possess a minimum. General stability conditions are derived, and it is shown that stable zones always exist on the axis of a field with rotational symmetry, and include the inflection point of the magnitude of the field. For the field inside a solenoid, the zone is calculated in detail; if the solenoid is long, the zone is centred on the top end, and its vertical extent is about half the radius of the solenoid. The theory explains recent experiments by Geim et al, in which a variety of objects (one of which was a living frog) was levitated in a field of about 16 T. Similar ideas explain the stability of a spinning magnet (Levitron™) above a magnetized base plate. Stable levitation of paramagnets is impossible.

1. Introduction

It is fascinating to see objects floating without material support or suspension. In the 1980s, this became a familiar sight when pellets of the new high-temperature type II superconductors were levitated above permanent magnets, and vice versa (Brandt 1989) (levitation of type I superconductors had been achieved much earlier (Arkadiev 1947, Shoenberg 1952)). Recently, two other kinds of magnetic levitation have captured the attention of physicists and the general public. In the Levitron TM (Berry 1996, Simon et al 1997, Jones et al 1997), a permanent magnet in the form of a spinning top floats above a fixed base that is also permanently magnetized. In diamagnetic levitation, recently achieved by A.K. Geim with J.C. Maan, H. Carmona and P. Main (Rodgers 1997), small objects (live frogs and grasshoppers, waterdrops, flowers, hazelnuts...) float in the large (16 T) magnetic field inside a solenoid. As well as being striking to the eye, magnetic levitation is particularly surprising to physicists because of the obstruction presented by Earnshaw’s theorem (Earnshaw 1842, Page and Adams 1958, Scott 1959). This states that no stationary object made of charges, magnets and masses in a fixed configuration can be held in stable equilibrium by any combination of static electric, magnetic or gravitational forces, that is, by any forces derivable from a potential satisfying Laplace’s equation. The proof is simple: the stable equilibrium of such an object would require its energy to possess a minimum, which is impossible because the energy must satisfy Laplace’s equation, whose solutions have no isolated minima (or maxima), only saddles.

Our purpose here is to explain how stable magnetic levitation of diamagnets can occur despite Earnshaw’s theorem. To do this, we obtain formulas for the energy and equilibrium of a diamagnet in magnetic and gravitational fields (sect. 2), and then derive the general conditions for the stability of the equilibrium (sect. 3). Stability is restricted to certain small zones, which we calculate in detail (sect. 4) for the field inside a solenoid. Finally, we describe (sect. 5) the diamagnetic levitation experiments carried out by Geim et al.

The explanation of the stability of the diamagnets is mathematically related to that of the Levitron™, but since the Levitron™ has already been treated in several papers we will restrict ourselves here to mentioning the similarities and differences between the two cases. We do not consider the levitation of high-temperature superconductors; this is stabilized by a different mechanism, involving dissipation (dry friction) caused by flux lines jumping between defects that pin them (Brandt 1990, Davis et al 1988). Nor do we discuss traps for microscopic particles, some of which are similar to the Levitron™ (Berry 1996) and some of which evade Earnshaw’s theorem through time-dependent fields (Paul 1990).

2. Energy and equilibrium

Let the magnetic field inside a vertical solenoid at position r = {x, y, z} be B(r) (fig. 1), with strength B(r) = |B(r)|, and let the gravitational field have acceleration g. The object that will be levitated in these fields has mass M, volume V (and density ρ = M/V), and magnetic susceptibility χ. For diamagnetic materials, χ < 0 (the special case χ = −1 corresponds to superconductors, i.e. perfect diamagnets), so we write χ = −|χ|.

For paramagnets χ > 0, but as we will show in sect. 3 levitation is impossible for these materials. We will be interested in substances for which |χ| « 1. Then, to a close approximation, the induced magnetic moment m(r) is

(1)

(In a more accurate treatment (Landau et al 1984), incorporating the distortion of the ambient field by the object, there is a shape-dependent correction to (1); for a sphere, the r.h.s. is divided by 1 − ^|χ|/₃. In general, the relation between B and M is tensorial.)

By integrating the work −dm · B as the field is increased from zero to B(r), we can obtain the total magnetic energy of the object and, adding this to the gravitational energy, the total energy:

(2)

For the object to be floating in equilibrium, the total force F(r) must vanish. Thus

(3)

where e_z is the upwards unit vector.

All the fields we are interested in will have rotation symmetry about e_z (continuous for a solenoid, discrete for the Levitron™ whose base is square). So, considering equilibria on the axis and denoting the field strength by B(z), the equilibrium condition becomes

(4)

Note that this involves only the density of the levitated object, not its mass.

For the Levitron™, the spinning-top is magnetized with magnetic moment m directed along the symmetry axis of the top. The purpose of the spin is to keep m gyroscopically oriented in the direction for which the force ∇m · B(r) from the base is upwards, that is, with m antiparallel to the effective dipole representing the base, since unlike dipoles repel (unlike unlike poles). Thus magnetic repulsion can balance gravity. (Without spin, the magnet orients itself parallel to the dipole representing the base, and is therefore attracted to the base, and falls.) The magnetic torque causes m to precess about the local direction of B(r). If this precession is fast enough (in comparison with the rate at which the direction of B(r) changes as the top bobs and weaves during its oscillations about equilibrium), a dynamical adiabatic theorem (Berry 1996) ensures that the angle between m and B(r) is preserved. For the Levitron™, m is approximately antiparallel to B(r), so this angle is close to 180°, and the energy is

(5)

Comparing (2) and (5), we see that the energy, and therefore the equilibrium, of both a diamagnetically levitated object and the Levitron™, depends on the magnitude B(r) of the field; at the end of sect. 3 we will see that this dependence is crucial to stability in both cases.

3. Stability

For levitation, the equilibrium must be stable, so that the energy must be a minimum, that is, the force F(r) must be restoring. We begin by showing that this excludes the levitation of paramagnetic objects. A necessary condition for stability is

(6)

where the integral is over any small closed surface surrounding the equilibrium point. From the divergence theorem, this implies ∇ · F(r) < 0, and hence, from (2) written for paramagnets, that is with |χ| replaced by −χ, that

∇2B2(r) < 0

(7)

But

(8)

where the last equality follows from the fact that the components of B satisfy Laplace’s equation (because there are no magnetic monopoles, so that ∇ · B = 0, and no currents within the solenoid, so that ∇ × B = 0). Therefore the necessary condition (6) for stability is violated, and stable levitation of paramagnets is impossible. That is why the equations in sect. 2 were written in the form appropriate for diamagnets.

Equation (8) is the essential step in the proof that the magnitude B(r) of a magnetic field in free space can possess a minimum but not a maximum. This theorem is

(and particularly by physicists who construct traps for microscopic particles) but we do not know who first proved it. It applies to any field that is divergenceless and irrotational. To a good approximation, it applies to velocity fields in the ocean, with the surprising consequence that there is no point within the Pacific Ocean where the water is flowing faster than at all neighbouring points; therefore places where the current has maximum speed lie on the surface.

The sufficient conditions for stability (as opposed to (6), which is merely necessary) are that the energy must increase in all directions from an equilibrium point satisfying (3), that is

(9)

For diamagnets, it now follows from (2) that

(vertical stability) (horizontal stability)

(10)

Because of the rotational symmetry, the last two conditions are equivalent. Now we show that the conditions can be conveniently expressed in terms of the magnetic field on the axis, B(z), and its derivatives B′(z) and B″(z). We begin by introducing the magnetic potential Φ(r), satisfying

B(r) = ∇Φ(r)

(11)

and its derivatives on the axis

(12)

From the fact that 8 satisfies Laplace’s equation, and rotational symmetry, there follows

(13)

Therefore the potential close to the axis can be written

(14)

From (11), the field strength can now be written

(15)

The stability conditions (10) can now be expressed in terms of φ_n(z), and thence in terms of the field on the axis:

(vertical stability) (horizontal stability)

(16)

For the Levitron™, where the magnetic energy (5) depends on B(r) rather than B²(r), a similar analysis leads to the same horizontal stability condition, and the simpler vertical stability condition B″(r) > 0. Mathematically, the reason why diamagnets and the Levitron™ can be levitated in spite of Earnshaw’s theorem is that the energy depends on the field strength B(r), which unlike any of its components does not satisfy Laplace’s equation and so can possess a minimum. Physically, the diamagnet violates the conditions of the theorem because its magnetization m is not fixed but depends on the field it is in, via (1). Microscopically, this is because diamagnetism originates in the orbital motion of electrons and so is dynamical. In the Levitron™, the magnitude of m is fixed but its direction is slaved to the direction of B(r) by an adiabatic mechanism that is also dynamical (at the macroscopic level) because it relies on the fast precession of the top. The (non-dissipative) stability of permanent magnets levitated above a (concave upwards) bowl-shaped base of type I superconductor (e.g. lead) (Arkadiev 1947) is similar to that of the diamagnets we have been considering. The superconductor is a perfect diamagnet (χ = −1), and so the permanent magnet above it is repelled by the field of the image it induces (Saslow 1991). If the magnet moves sideways, the image gets closer, so that the energy increases.