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

4. Stable zone

On the axis of a solenoid, or above the base of a Levitron™, the field B(z) decreases monotonically as z increases from 0 to ∞, and there is an inflection point at some height z_i, that is B″(z_i) = 0. At z_i, both discriminants D₁ and D₂ in (16) are obviously positive, so the equilibrium is stable at z_i . Simple geometrical arguments show that D₁ has a zero at a point z₁ < z_i, and vertical stability requires z > z₁; similarly, D₂ has a zero at a point z₂ > z_i, and horizontal stability requires z < z₂. This establishes the existence of a stable zone on the axis, namely z₁ < z < z₂, within which diamagnetic objects can be levitated.

It is necessary for the equilibrium position satisfying (4) to lie in the stable zone. This can be achieved by changing the current in the solenoid, which scales the magnetic field strength B(r) while preserving the geometry of the field lines and therefore the stable zone determined by (16).

In the Levitron™, the stable zone is z_i < z < z₂, and, since the base is a permanent magnet whose field cannot easily be altered, the equilibrium height of the floating top can be brought into this interval by adding or removing small washers to change the weight Mg.

As a model to study in detail, we consider the field inside a long solenoid of length L and radius a (fig. 1). Then, defining the scaled variables

(17)

and the field B₀ at the centre of the solenoid z = 0, we have, introducing obvious notations,

(18)

There are inflections close to the ends ζ = ±¹/₂ of the solenoid; levitation occurs near the top end, that is ζ = +¹/₂, where the field gradient is negative as required by (4).

Fig. 2. (a) Field on the axis inside a solenoid with δ = ^2a/_L = 0.1;
(b) the discriminants D₁(ζ) and D₂(ζ) defined by (16), and the stable zone where both are positive.

Fig. 2 illustrates this field, and the corresponding discriminants (16), for δ = 0.1. The stable zone is ζ₁ = 0.487083 < ζ < ζ₂ = 0.510223.

For thin solenoids (δ « 1), some simplification is possible, since then the second term in (18) can be approximated by unity near ζ = ¹/₂. A short analysis shows that in this limit the inflection and stable zone are, when expressed in the original z coordinate,

z1 = 1/2L z1 = 1/2L - 0.258199a < z < z2 = 1/2L + 0.204124a (L » a)

(19)

For fat solenoids (δ » 1), simplification is again possible, because then the field is that on the axis of a current loop, namely

(a » L)

(20)

From (16), the inflection and stable zone are

(a » L)

(21)

By Ampère’s equivalence between distributions of magnetization and current loops, the field (20) is the same as that on the axis of a uniformly magnetized disc. Therefore, with the vertical stability condition B″(r) > 0 (see the remark following equation (16)), (21) leads to the stable zone previously calculated (Berry 1996) for a Levitron™ with a circular disc base, namely (If the base of the Levitron™ is a ring, rather than a disc, the stable region is much higher, namely 1.6939a < z < 1.8253a, and this explains the operation of the recently developed ‘superlevitron’.) It is instructive to display spatial contour maps of the energy (2) as the field B₀ at the centre of the solenoid is varied, showing the appearance and disappearance of the minimum as the equilibrium enters and leaves the stable zone. We employ the dimensionless field β and energy E defined by

(22)

where, in terms of (15) and the field profile (18),

(23)

(the primes denote ∂/∂ζ). From the equilibrium condition (4), the field β(z) for which the diamagnet floats at height ζ is

(24)

Fig. 3. Contours of the scaled energy E(ξ, η, ζ; β, δ) (gravitational + magnetic) for a diamagnet, for different values of the dimensionless field β (defined by (22)) at the centre of a solenoid with δ = 0.1.
(a) β = 0.527046; (b) β = β₂ = 0.513563, i.e. levitation at z₂ ; (c) β = 0.445301, i.e. levitation at z_i; (d) β = β₁ = 0.417998; (e) β = 0.411693, i.e. levitation at z₁

Fig. 3 shows the E landscape as the field β is decreased through the stable range, for a solenoid with δ = 0.1. At the top of the range (fig. 3(b) β = β₂ = 0.513563, corresponding to equilibrium at the upper limit z = z₂ of the stable zone, and at the bottom of the range (fig. 3(d)) β = β₁ = 0.417998, corresponding to equilibrium at the lower limit z = z₁ of the stable zone. At β₂ the minimum is born (along with two off-axis saddles) from the splitting of an axial saddle; at β₁, the minimum dies as it annihilates with another axial saddle. We caution against quantitative reliance on the details of these landscapes near the wall of the solenoid (e.g. near ξ = 0.05 in fig. 3), because they are based on the quadratic approximation (23), which is strictly valid only close to the axis. Stably levitated diamagnets can make small, approximately harmonic, oscillations near the energy minimum, and these are observed as the gentle bobbing and weaving of the objects. Larger oscillations will be anharmonic. The region they explore has the form of a conical pocket (fig. 3(c)), in which motion is almost certainly nonintegrable and probably chaotic. We think this would repay further study, but here confine ourselves to estimating the greatest lateral extent of the region in which the oscillations occur. From fig. 3, it is reasonable to define this as the distance from the axis to the off-axis saddles for the field that corresponds to equilibrium at z_i, namely β = 0.445301. It follows from (23) that these saddles lie at z = z₂, and use of (4) then leads to

(25)

For thin solenoids, this can be evaluated as

R = 0.75569a (L ≫ a).

(26)

When δ = 0.1 this gives ^R/_L = 0.0377, in agreement with fig. 3(c) (which was calculated without the thin- solenoid approximation).

5. Experiment

Most diamagnetic materials have susceptibilities of order χ ≈ −10⁻⁵. For water, χ = −8.8 × 10⁻⁶ (Kaye and Laby 1973), and using ρ = 1000 kgm⁻³ the equilibrium condition (4) gives the required product of field and field gradient as

B(z)B′(z) = -1400.9T2m-1

(27)

This has been achieved in experiments involving one of us (Geim et al) with a Bitter magnet whose geometry is shown in fig. 4(a). The operation of this electromagnet consumed 4 MW, but we emphasize that this is power dissipated in the coils, not power required for levitation - indeed, with the field of a persistent current in a superconducting magnet levitation can be maintained without supplying any energy.

The measured field profile is shown in fig. 5. The inflection point is at z_i = 78 mm, where the field is B(z_i) = 0.63B₀ and the gradient of the field at z_i is −8.15B₀ T m⁻¹, from which the required central field is predicted via (27) to be

B0 = 16.5T.

(28)

From the measured data we have calculated the discriminants D₁ and D₂ defined by (16), and thence the stable zone, which is predicted to be z₁ = 67.5 mm < z < z₂ = 87.5 mm.

Fig. 4(a). Geometry of coils in Bitter magnet used for levitating diamagnetic objects. The currents in the two coils were equal. The region of stable levitation is near the top of coil 1, and marked with a dot.

Fig. 4(b). Frog levitated in the stable region.

A variety of diamagnetic objects was inserted into the magnet, and the current through the coils adjusted until stable levitation occurred (fig. 4(b)). The corresponding fields B₀ were all close to the calculated 16 T, and the objects always floated near the top of the inner coil, as predicted. Careful observation of a (3 mm diameter) plastic sphere showed that it could be held stably in the range (69 ± 1) mm < z < (86 ± 1) mm, in very good agreement with theory.

The induced dipole m (equation (1)) responsible for the levitation of a diamagnet can be regarded as equivalent to a current I = ^|m|/_A circulating in a loop of area A embracing it. For an object of radius 10 mm, such as the very young frog that was levitated (fig. 4(b)), this current is about 1.5 A (corresponding to a field B ≈ 10⁻⁵ B₀ ≈ 1.5 Gauss induced inside the frog). Of course this represents the summation of microscopic currents localized in atoms, not the bulk transport of charge, so the living creatures were not electrocuted. Indeed, they emerged from their ordeal in the solenoid without suffering any noticeable biological effects - see also Schenck (1992) and Kanal (1996).

As we showed earlier, it is impossible to levitate paramagnets stably. Balance of forces can however be achieved, and from (4) with the sign reversed it is clear that this occurs for z < 0, and close to the centre of the solenoid - rather than near the bottom - because χ_paramagnetic ≈ 10⁻³ ≈ 100χ_diamagnetic; this position is vertically stable but laterally unstable. Nevertheless, some paramagnetic objects (Al, several types of brass, stainless steel, paramagnetic salts with Mn and Cu) were suspended in this way, but not levitated: they were held against the side wall of the inner coil. On a few occasions, paramagnets floated without apparent contact, but were found to be buoyed up by a rising current of paramagnetic air; when this was inhibited, for example by covering the ends of the solenoid with gauze, the objects slipped sideways and were again held against the wall.

Fig. 5. Profile of field on axis of Bitter magnet in fig. 4, measured at intervals of 10 mm, showing the stable zone near the top of coil 1.

6. Discussion

Our treatment of diamagnetic levitation has neglected at least three small effects that could have interesting consequences. The first arises from the shape-dependence of the induced magnetic moment. For living organisms (e.g. frogs) trapped in the energy minimum this could be exploited to provide an escape mechanism. If the frog is initially in equilibrium, there are no forces on it. By changing shape (e.g. from a sphere to an ellipsoid) the induced moment will change (Landau et al 1984), and the force will no longer be zero, so the frog will start to oscillate about a slightly different point. By repeating this manoeuvre at the frequency of oscillations in the minimum, the oscillations will be amplified by parametric resonance until the frog leaves the stable zone. This is a tiny effect, because the shape-dependence of m is of the order |χ| ≈ 10⁻⁵, so escape would require 10⁵ such ‘swimming strokes’; therefore the frog would have to be persistent as well as highly coordinated. (In practice, the frog does try to swim - but in the ordinary way, by paddling the air in the solenoid - but nevertheless remains held in the energy minimum, for the entire observation - up to 30 minutes.)

The second effect arises from the finite extent of any real levitated object. Its equilibrium depends on the total magnetic force, which must balance the weight. The local force balance (4) will occur only at one height z_b in the body. For z < z_b, the net force on each element will be upwards, and for z > z_b the net force will be downwards. Therefore the object will be compressed to an extent that depends on how much BB′ varies across it, that is on the curvature of B²(z) at z_b. A land-based living creature would be unlikely to feel this effect, since it is already accustomed to a much greater inhomogeneity: the external upward force that balances gravity is concentrated in a molecular layer in the soles of its feet.

The third effect occurs for objects that are diamagnetically inhomogeneous, so that their different parts (e.g. flesh and bone for a living organism) have different χs. Then, as just described for an extended object, the force balance will be different at different points. This could cause strange sensations; for example, if |χ|_flesh > |χ|_bone the creature would be suspended by its flesh with its bones hanging down inside, in a bizarre reversal of the usual situation that could inspire a new (and expensive) type of face-lift (since |χ|_bone ≈ |χ|_water (Schenck 1992) this would require |χ|_flesh > |χ|_water).

Acknowledgement

AKG thanks the staff at the High Field Magnet Laboratory (University of Nijmegen) for technical assistance, and the European Community Program ’Access to Large Scale Facilities’ for financial support.

References