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Unformatted text preview: PII: S01430807(97)846892 Eur. J. Phys. 18 (1997) 307–313. Printed in the UK Of ﬂying frogs and levitrons
M V Berry† and A K Geim‡
† H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK
‡ High Field Magnet Laboratory, Department of Physics, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen,
The Netherlands
Received 4 June 1997
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
(LevitronTM ) above a magnetized base plate. Stable levitation
of paramagnets is impossible. Samenvatting. Magnetische velden stoten diamagnetische
voorwerpen af. Zulke velden kunnen zo sterk zijn dat zij de
zwaartekracht opheffen. Het is op deze wijze mogelijk zulke
voorwerpen te laten zweven. Dit vormt een stabiel
evenwicht, wat in tegenspraak schijnt te zijn met Earnshaw’s
Theorema. Echter Earnshaw’s Theorema is niet langer geldig
als het magnetisme veld geinduceerd is. De totale energie
(bevattende bijdragen van het magnetisme en de
zwaartekracht) kan toch een lokaal minimum vertonen.
Algemene criteria voor zo’n minimum zullen worden
opgesteld. Verder zal worden aangetoond dat voor een
cilindrisch symmetrisch veld, langs zijn symmetrie as altijd
een zone gevonden kan worden waarin een stabiel evenwicht
bestaat. Voor het veld binnen een soleno¨de zal deze zone in
ı
detail bepaald worden. Als deze spoel voldoende land is
bevindt deze zone zich aan het uiteinde van de spoel. De
lengte van deze zone langs de symmetrie as van het veld is
ongeveer de helft van de straal van de spoel. Deze theorie
geeft een goede verklaring voor de experimenten van Geim et
al. In deze experimenten werden een grote verscheidenheid
aan verschillende voorwerpen (waaronder een levende kikker)
tot zweven gebracht in velden van ongeveer 16 T. Analoge
theorien verklaren de stabiliteit van een roterend permanent
¨
magneetje (LevitronTM ) boven een magneetische grondplaat.
Het is onmogelijk om paramagnetische voorwerpen stabiel te
doen zweven. 1. Introduction (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 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 hightemperature
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 LevitronTM
(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 c
01430807/97/040307+07$19.50 1997 IOP Publishing Ltd & The European Physical Society 307 308 M V Berry and A K Geim energy and equilibrium of a diamagnet in magnetic
and gravitational ﬁelds (section 2), and then derive the
general conditions for the stability of the equilibrium
(section 3). Stability is restricted to certain small zones,
which we calculate in detail (section 4) for the ﬁeld
inside a solenoid. Finally, we describe (section 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 LevitronTM ,
but since the LevitronTM 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 hightemperature 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 LevitronTM (Berry 1996) and some
of which evade Earnshaw’s theorem through timedependent ﬁelds (Paul 1990).
2. Energy and equilibrium
Let the magnetic ﬁeld inside a vertical solenoid at
position r = {x, y, z } be B (r ) (ﬁgure 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 section 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
m (r ) = − χ V B (r )
.
µ0 (1) (In a more accurate treatment (Landau et al 1984),
incorporating the distortion of the ambient ﬁeld by the
object, there is a shapedependent 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:
E(r ) = mgz + χ V 2
B (r ).
2µ0 (2) For the object to be ﬂoating in equilibrium, the total
force F (r ) must vanish. Thus
F (r ) = −∇ E(r ) = −mg ez − χ V
B(r )∇ B(r ) = 0
µ0
(3) Figure 1. Geometry and notation for ﬁeld in a solenoid. where ez is the upwards unit vector. All the ﬁelds we
are interested in will have rotation symmetry about ez
(continuous for a solenoid, discrete for the LevitronTM
whose base is square). So, considering equilibria on
the axis and denoting the ﬁeld strength by B(z), the
equilibrium condition becomes
µ0 ρ g
B(z)B (z) = −
.
(4)
χ 
Note that this involves only the density of the levitated
object, not its mass.
For the LevitronTM , the spinningtop 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
LevitronTM , m is approximately antiparallel to B (r ),
so this angle is close to 180◦ , and the energy is
E(r ) = mgz − m · B (r ) ≈ mgz + mB(r ). (5) Comparing (2) and (5), we see that the energy, and
therefore the equilibrium, of both a diamagnetically
levitated object and the LevitronTM , depends on the
magnitude B(r ) of the ﬁeld; at the end of section 3
we will see that this dependence is crucial to stability
in both cases. Diamagnetic levitation 309 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
F (r ) · dS < 0
(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 But ∇ 2 B 2 (r ) < 0. (7) = φ0 (z) − 1 (x 2 + y 2 )φ2 (z) + . . . .
4
From (11), the ﬁeld strength can now be written
2
B 2 (r ) = φ1 (z) + 1 (x 2 + y 2 )
4 2
2
2
∇ 2 B 2 (r ) = ∇ 2 B x + B y + B z
2
= 2 ∇ Bx 2 + ∇ By + ∇ Bz 2
+Bx ∇ 2 Bx + By ∇ 2 By + Bz ∇ 2 Bz
2
= 2 ∇ Bx 2 + ∇ By + ∇ Bz 2 ≥ 0 (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 section 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 ‘well known to those who know well’
(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
2
∂x E(r ) > 0 2
∂y E(r ) > 0 2
∂z E(r ) > 0. 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 (r ),
satisfying
B (r ) = ∇ (r )
(11)
and its derivatives on the axis
n
φn (z) ≡ ∂z (0, 0, z).
(12)
From the fact that satisﬁes Laplace’s equation, and
rotational symmetry, there follows
2
2
∂x (0, 0, z) = ∂y (0, 0, z) = − 1 φ2 (z).
(13)
2
Therefore the potential close to the axis can be written
2
(r ) = φ0 (z) + 1 x 2 ∂x (0, 0, z)
2
2
+y 2 ∂y (0, 0, z) + . . . (9) For diamagnets, it now follows from (2) that
2
∂z B 2 (r ) > 0 (vertical stability)
2
2
∂x B 2 (r ) > 0 ∂y B 2 (r ) > 0 (horizontal stability).
(10) Because of the rotational symmetry, the last two
conditions are equivalent. Now we show that the (14) 2
× (φ2 (z) − 2φ1 (z)φ3 (z)) + . . . .
(15)
The stability conditions (10) can now be expressed in
terms of φn (z), and thence in terms of the ﬁeld on the
axis:
D1 (z) ≡ B (z)2 + B(z)B (z) > 0 (vertical stability)
D2 (z) ≡ B (z) − 2B(z)B (z) > 0
2 (16) (horizontal stability).
For the LevitronTM , where the magnetic energy (5)
depends on B(r ) rather than B 2 (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
LevitronTM 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 LevitronTM , 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 (nondissipative) stability of permanent magnets
levitated above a (concave upwards) bowlshaped 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. 310 M V Berry and A K Geim 4. Stable zones
On the axis of a solenoid, or above the base of a
LevitronTM , the ﬁeld B(z) decreases monotonically as
z increases from 0 to ∞, and there is an inﬂection
point at some height zi , that is B (zi ) = 0. At zi , both
discriminants D1 and D2 in (16) are obviously positive,
so the equilibrium is stable at zi . Simple geometrical
arguments show that D1 has a zero at a point z1 < zi ,
and vertical stability requires z > z1 ; similarly, D2 has a
zero at a point z2 > zi , 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 LevitronTM , the stable zone is zi < 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 (ﬁgure
1). Then, deﬁning the scaled variables
ξ ≡ x/a, η ≡ y/a,
ζ ≡ z/L and δ ≡ 2a/L
(17)
and the ﬁeld B0 at the centre of the solenoid z = 0, we
have, introducing obvious notations,
B(ζ , δ )
≡ B(ζ , δ ) = 1 1 + δ 2
2
B0
1 − 2ζ
1 + 2ζ
×
+
.
(1 − 2ζ )2 + δ 2
(1 + 2ζ )2 + δ 2
(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). Figure 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,
zi = 1 L
2
z1 = 1 L − 0.258199a < z < z2 = 1 L + 0.204124a
2
2 (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
B0
(a L).
(20)
B(z) =
(1 + (z/a)2 )3/2 Figure 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. From (16), the inﬂection and stable zone are
zi = 1 a
2
z1 = 1
√a
7 = 0.378a < z < z2 = (a L). 2
a
5 = 0.6325a (21) By Ampere’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 LevitronTM with a circular disc base, namely
√
a/2 < z < a (2/5). (If the base of the LevitronTM 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
2
B0 ≡ β 2 E(r ) ≡ ρ gLµ0
χ  2
χ V B0
E(ξ , η, ζ ; β , δ )
2µ0 (22) Diamagnetic levitation 311 where, in terms of (15) and the ﬁeld proﬁle (18),
E (ξ , η , ζ ; β , δ ) ≡
2
ζ + 1 B(ζ , δ )2 + 1 (ξ 2 + η2 )
4
4
β2
2
× {B (ζ , δ ) − 2B(ζ , δ )B (ζ , δ )}
(23) (the primes denote ∂ /∂ζ ).
From the equilibrium
condition (4), the ﬁeld β (z) for which the diamagnet
ﬂoats at height ζ is
−1
(24)
β (z)2 = − B(ζ , δ )B (ζ , δ ) . Figure 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 (ﬁgure 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 (ﬁgure 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 offaxis 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 ﬁgure 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 (ﬁgure 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 ﬁgure 3, it is rea
sonable to deﬁne this as the distance R = (x 2 + y 2 )
from the axis to the offaxis saddles for the ﬁeld that
corresponds to equilibrium at zi , namely β = 0.445301.
It follows from (23) that these saddles lie at z = z2 , and
use of (4) then leads to
B(ζi , δ )B (ζi , δ ) − B(ζ2 , δ )B (ζ2 , δ )
2
2
. (25)
R = 4L
B(ζ2 , δ )B (ζ2 , δ )
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 ﬁgure 3(c) (which was calculated without the thinsolenoid 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 kg m−3 Figure 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 zi ;
(d ) β = β1 = 0.417998;
(e) β = 0.411693, i.e. levitation at z1 . the equilibrium condition (4) gives the required product
of ﬁeld and ﬁeld gradient as
−1 B(z)B (z) = −1400.9 T2 m . (27) B0 = 16.5 T. (28) This has been achieved in experiments involving one of
us (Geim et al) with a Bitter magnet whose geometry
is shown in ﬁgure 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 ﬁgure 5. The
inﬂection point is at zi = 78 mm, where the ﬁeld is
B(zi ) = 0.63B0 and the gradient of the ﬁeld at zi is
−8.15B0 T m−1 , from which the required central ﬁeld
is predicted via (27) to be
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. 312 M V Berry and A K Geim Figure 5. Proﬁle of ﬁeld on axis of Bitter magnet in
ﬁgure 4, measured at intervals of 10 mm, showing the
stable zone near the top of coil 1.
Figure 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. Figure 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 (ﬁgure 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 (ﬁgure
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
χparamagnetic ≈ 10−3 ≈ 100χdiamagnetic ; 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.
6. Discussion
Our treatment of diamagnetic levitation has neglected
at least three small effects that could have interesting
consequences.
The ﬁrst arises from the shapedependence 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 Diamagnetic levitation
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 shapedependence 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 zb
in the body. For z < zb , the net force on each element
will be upwards, and for z > zb 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 B 2 (z) at zb . A
landbased 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 χ s. 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 facelift
(since χ bone ≈ χ water (Schenck 1992) this would
require χ ﬂesh > χ water ). 313
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.
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This note was uploaded on 12/29/2011 for the course PHYSICS 411 taught by Professor Agshe during the Summer '11 term at Maryland.
 Summer '11
 Agshe
 Physics, Magnetism

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