*A Toy
Story*

The Chemical Relevance of

Earnshaw's Theorem, and

How the
Levitron^{®}
Circumvents It

*J. M. McBride*

Department of Chemistry

Yale University

*Poster presented at*

*Gordon Research Conference on
Physical-Organic Chemistry*

*June 29-July 4, 1997*

E-mail j.mcbride@yale.edu

© 1997 J.M. McBride

last modified August 27, 2001

*CONTENTS*

1.**
Earnshaw's Theorem**

6.
**How does Earnshaw Impact
****Chemistry****?**

9.
**What is the Levitron?**

11.

13.

15.

19.

20.

21.

**1.****
Earnshaw's Theorem**

On March 18, **1839** at the Cambridge
Philosophical Society **Samuel
Earnshaw** of St. John's College read
his paper:

Earnshaw questions whether the restoring force
that allows particles of the ether to vibrate, as they propagate a
light wave, might obey an **inverse square law**, like gravity. He
decides no, based on a proof that depends on the divergence of such a
force being zero:

**Del ^{.}F = dF / dx +
dF / dy + dF / dz = 0 **

Since they sum to zero, the restoring forces in
three orthogonal directions cannot all have the same sign.
**There can be no local
minimum (or maximum) of energy for an inverse square force
law.** So vibration about a point of stable
equilibrium is impossible with such a force. Q.E.D.

2.

Of course Coulomb's force obeys an inverse square
law, so there can be no position of stable equilibrium for a
**charge**
in a region free of opposing charges.

[A graphic proof is easily to visualize. Lines
of force would have to converge on any position where a charged
particle is to be in stable equilibrium, but they can converge only
on a charge of the *opposite* sign.]

**Gravitation**
and **magnetic
poles** also obey inverse square force
laws.

Since each of these forces alone has a divergence of zero, so must the total force resulting from any combination of them.

3.

(or maximum) of potential energy

for any set of bodies interacting by

gravitation, electrostatics,

and/or magnetostatics.

[The only critical points are saddles.]

which do not obey inverse square forces?

There *can* be local energy minima for
diamagnets (like frogs)
and superconductors, but
Earnshaw's Theorem forbids energy minima for permanent
multipoles, even though they obey
r^{-}* ^{n}*
forces with

[For multipoles with *fixed* orientation
the divergence of component pole-pole forces is zero. If one allows
spontaneous reorientation, it becomes even easier to escape from a
putative local minimum.]

There *can* exist local minima (but not
maxima) for the ** total** field strength, but there can be
neither minima nor maxima for any

If there were a local maximum of total field, a
favorably oriented dipole could be trapped there, but *there are no
local maxima*.

At a local minimum, an *un*favorably oriented
dipole could be trapped, but it would flop over to become favorable.

When the electron was discovered (100 years ago, August 1897), Earnshaw's Theorem gave physicists headaches.

To satisfy Earnshaw with a static atom, J. J. Thompson proposed a "plum pudding" model: he planted electrons within a diffuse positive charge.

Thompson also used magnetized pins stuck in
floating corks to rationalize the magic numbers of the periodic
table. [J.J. Thomson, *The
Corpuscular Theory of Matter*, Scribners, New York, 1907, p.
110ff.]

He had no way to know that in fact kinetic energy
makes the *electrons* diffuse, so that nuclei are trapped at
energy minima in the midst of a time-averaged electron
cloud.

Chemist G. N. Lewis (like physicist James Jeans) suggested that Coulomb's Law might break down in order for atoms to be stable at rest while satisfying Earnshaw's Theorem.

"...if we use the electron as a test charge to determine the properties of the simplest possible electric field, namely the field about a hydrogen nucleus, we appear to find that this field is not a continuum but is strikingly discontinuous."

"Instead of thinking then of an electric field as a continuum, we should rather regard it as an intensely complicated mesh..."

[It now seems that although Coulomb's Law
fails by the Planck

length
(10^{-}* ^{35}*
m), it holds at all chemically relevant
distances.]

e.g. Question:

above an

(Neglect discreteness of the dipoles. Assume continuous sheets of charge.)

Which way is the positive charge pushed? the
negative charge? the dipole?

Is the dipole optimally oriented?

"*Levitron*" is the tradename of an amazing
levitating top. (For examples click to visit levitron.com)

Roy Harrigan (Manchester, VT) counts the "Levitation Device" as one among more than 100,000 of his inventions. He built it in the late 1970's and received U.S. Patent 4,382,245 on May 3, 1983. In 1993 Harrigan gave a working prototype
to Bill Hones (Fascinations, Inc. Seattle), in expectation
of establishing a partnership for commercialization. Five
months later Hones applied for his
own separate patent, which issued
on April 4, 1995. It has been estimated that Fascinations,
Inc., has since sold, directly or under license, some
1,000,000 |
Roy Harrigan |

Knowledgeable physicists repeatedly told Harrigan that he was wasting his time because Earnshaw's Theorem forbids levitation of permanent magnets.

The
**conditions**
the Levitron must satisfy are almost impossibly stringent.

the** magnet
strengths** must be right;

the** mass of the
top** must be correct within 0.5%;

the** spinning
rate** must be neither too fast nor too
slow;

the top must be introduced into a
**small region of
stability**, only millimeters wide and high,
with

**tiny translational
energy** (less than required to escape a
dimple 50 microns deep).

Roy first achieved levitation spinning the top by hand in the air!

He tried spinning so often that now he catches falling objects "before I know they fell."

To a first approximation the spinning magnetic top acts as a gyroscope that stays oriented vertically in opposition to the field of the magnetic base.

But it violates Earnshaw's Theorem.

The cognoscenti know there must be more to it.

The full explanation was published by last summer by Michael Berry (Theoretical Physics, Bristol).

And independently by Simon, Heflinger, and Ridgway (Physics, UCLA)

Link to Simon et al.

The Simon *et al*. paper is particularly
approachable for someone as mathematically challenged as I am. Both
papers are important souces for this poster and present most of the
information with more rigor and detail. The poster intends to serve
as a qualitative appetizer to the Berry and the Simon *et al.*
papers by providing pictures to illustrate the conditions for
vertical and horizontal stability.

The force from an infinite base would be zero. At
great distance the force from a finite base would be proportional to
1/r^{4} (as for
point dipoles).

As an inverted vertical top falls toward a disk-shaped base on axis, the force opposing gravity increases to a maximum then falls to zero, as the base begins to look effectively infinite. Force is maximum at a height equal to half the base radius.

at height = 1/2 disk radius of the base.

A top of small enough mass will float at an appropriate level above the height of maximum force where vertical repulsion balances gravity.

Consider the circular "footprint" of a cone originating on the top and tangent to the disk of the base magnet when the top is on the axis of the base.

If the top drifts left, its footprint loses a small crescent from the rim of the base magnet. But the top gains an equivalent crescent of the base magnet at the right, further from the top's axis and outside the original footprint.

that are at a radius equal to twice the dipole's height

Base elements near the top's axis raise the energy. Those further from the top's axis lower it (to balance those closer in; remember, an infinite base gives no force). Beyond a certain distance the contribution decreases with increasing distance (the most favorable elements are at a distance from the top axis equal to half the top's altitude). Note that radial distance is measured in units of the height of the top.

17.

When the top is high above the base, the cone is narrow, that is, the base magnet is small measured in units of the top altitude. As shown in the graph below, the footprint is well within the steeply descending portion of the energy. When the top drifts to the left relative to the base, the crescent gained is less unfavorable that the crescent lost, and the net energy is lowered by moving off-axis.

High above the base, the top experiences a destabilizing horizontal force.

A lower top would make the base effectively larger
and would thus experience less unfavorable (or even favorable) energy
contributions from elements near the rim of the base. If the top is
low enough that the rim of the "enlarged" base extends beyond the
minima at -/+ 2, there is a small horizontal restoring force on the
top, because the net energy *increases* as the top shifts
off-axis.

Near the base, the inverted vertical top experiences a stabilizing horizontal force.

The border between regimes of horizontal stability and instability comes at the same height as the border between regimes of vertical instability and stability (half the base radius, so that the rim of the base magnet is at +/-2 in the plot above, where an element's contribution to top energy is most favorable).

An inverted top that is held
strictly vertical by

gyroscopic force has no region
of overall stability.

Earnshaw's Theorem seems to forbid success of the
*Levitron** ^{®}*.

The spinning top is not held strictly vertical. It can precess about the magnetic field direction at a small angle.

As the precessing top moves off-axis, it can tilt
slightly to follow the tilted local field.
Since** the total local field
is larger than its vertical component**, the
tilted top experiences a small increase in total energy that
increases the horizontal restoring force.

For a tilting top, horizonal stability disappears slightly above the height where vertical stability begins. Thus, there is a tiny region of absolute stability. The mass of the top must be large enough to reach vertical equilibrium below the upper limit for horizontal stability.

If the top spins too slowly, the gyroscopic torque is too small to keep it from flopping over and falling to the base.

The top can also spin too fast for stability. The precession rate is inversely proportional to spinning rate, so a rapidly spinning top does not track the local field as the top drifts off-axis. A too-fast top maintains its orientation, and Earnshaw's Theorem assures its instability.

Fortunately for Roy Harrigan, and for us, the range of stable spin rates (18-40 rps) is accessible by hand spinning.

**Theory:**

Michael Berry (Physics, Bristol)

Martin Simon (Physics, UCLA)

Charles Summerfield (Physics, Yale)

**History:**

Roy Harrigan (Inventor, Manchester, VT)

Martin Simon (Physics, UCLA)

**Toy and
Tolerance:**

Florence McBride

J. Michael McBride

`Department of
Chemistry`, Yale University

Box 208107, New Haven, CT 06520-8107

e-mail: j.mcbride@yale.edu