[Phys-L] levitation (was: Meisner Effect)

[John Denker <jsd@AV8N.COM>]

On 07/05/05 11:37, David Abineri wrote:
> I am finding it very difficult to explain the Meisner effect to my high
> school students and I wonder if someone can help to put it into their terms.

If you're having trouble explaining the Meissner effect
in terms of classical physics, that's a good sign;  it
means you're paying attention.   It is a completely
nonclassical quantum effect.

> Explanations that I have read talk about "expelling the field"  or
> "preventing penetration of the field" by the superconducting material

That's not an explanation of the Meissner effect.
That is perhaps a description of the effect.

> but I have a difficult time seeing this as the source of an upward force
> on the levitating magnet.

If you accept the Meissner effect as a fait accompli,
levitation can be understood as a consequence thereof.

In particular:  A superconductor is more than a perfect
classical conductor ... but a merely classical perfect
conductor suffices for levitation, if you initialize
it with zero trapped flux.  Perfect conduction implies
(via Lenz's law) perfect diamagnetism.

I changed the Subject: line, because I am assuming this
is a question about levitation, not a question about the
Meissner effect per se.  (If I guessed wrong, please
re-ask the question.)  Specifically:

    Meissner ==>  perfect conductor ==>
             diamagnetism ==>  levitation

and all we really need are the last two ideas, namely
   diamagnetism ==> levitation

Assume the diamagnet starts out far away, then gradually
move it into the field.  The magnetic field will not
penetrate the diamagnet.  Analyze the result as a
superposition of the original field plus the field
generated by the eddy currents.  The resulting field
has higher energy than the original field.  You can
verify this in the following simple case:  let the
original field be locally uniform.  Model the field of
the diamagnet as having the same pattern as a magnetic
dipole.  Calculate the field energy by integrating B^2
of the volume.

You have to be careful with this ... in particular you
must *not* try to model the diamagnet as a solenoid
with constant field inside and "zero" field outside.
It turns out that the field inside the solenoid, and
elsewhere along the axis of the solenoid, makes a negative
contribution to the energy, and the positive contribution
comes from the field outside, which is not really zero,
just small, and spreads over a huge volume.  This small
effect, integrated over a sufficiently huge volume,
produces the actual positive energy.

Finally, given this positive energy, argue by the
principle of virtual work that moving the perfect
diamagnet into original field must have involved, at
some point, a force pushing the diamagnet away from
the high-field region.

It is possible to arrive at the same result by a different
line of argument, using the Lorentz force law rather
than energy arguments, but it is tricky.

> And why is this upward force able to keep the
> levitating magnet in balance so well since, if I tried to levitate one
> magnet with another, I would have a very difficult time keeping it in
> equilibrium.

Difficult indeed.  Earnshaw's theorem says there is no
equilibrium involving permanent magnets.

The loophole is that a diamagnet is not a permanent
magnet;  specifically, it does not satisfy the premises
of Earnshaw's theorem.  If you move the diagmagnet
around in a nonuniform applied field, its degree of
magnetization varies.

To say the same thing another way, the energy of a
permanent magnet in an applied field is linear in the
strength of the applied field, whereas the energy of
a diagmagnet is quadratic.  That nonlinearity makes
a big difference.
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