W9_parallel_resonance.eps

294 Week 8: Faraday’s Law and Induction

surface current

internal atomic (spin) currents

N

S

B inside

Figure 111: In a ferromagnet, the magnetic dipoles spontaneouslyalign when cooled below a critical
temperature. The resulting surface current transforms them into small “solenoids” with a non-
dissipative surface current surrounding their interior volume and trapping magnetic flux that emerges
from their north pole and flows to their south pole.

excellent example of the (sudden) emergence oflong range orderin a system that is disordered at
high temperatures. The magnetic susceptibility of the system, theheat capacity of the system, and
other thermodynamic descriptors of the system all do unusual things at the critical temperature of
the phase transition, often exhibiting divergent or non-continuous behavior. Considerable effort has
been expended on deriving a theory that accurately describes things like the particular value of the
critical temperature and certain exponents that describe the divergences that occur there. These
theories haven’t been without some successes, but only a very fewsimple models have been solved
exactly, notably the 2 dimensional Ising model mentioned and portrayed in cartoon form above.

However, we can now use powerful computers to simulate the behavior of “ideal” magnetic
systems and compute their critical parameters with systematicallyimprovable accuracy. These
computations in turn can be used to check the theoretical predictions (since we lack “perfect”
exemplars of the theoretical models in messy old nature).

Magnetism, Concluded

With this we’ll wrap up our treatment of straight-up magnetic phenomena. As you can see, it is
considerably more complicated than electrostatics even before the dynamical behavior associated
with Faraday’s Law is introduced.

Magnetic forces are right-hand twisty. They appear to violate Newton’s Third Law, whichshould
make you very worried about the consistency of physics and the laws of Conservation of Momentum
and Angular Momentum. They appear or disappear, seeming to turnsomehow into the electric
force as we change inertial reference frames (transforming intoa frame where a charge is at rest, for
example).

The sources of magnetic fields are no less right-handed twisty. Fields circulate around moving
electriccharges, and although we might expect to find free magnetic charges, so far nobody has
managed to salt the tail of one^82.

(^82) Sorry, this is an ancient metaphor, associated with the ideathat you can catch a bird by putting salt on its tail.
It is used by bored parents to torment their four year old children who want to catch the pretty birdies. As in: “Oh,
you want to catch that sparrow? All you have to do is put salt onits tail!” The child, of course, spends days in the
field with a box of salt, trying to get close to birds. Birds, not beingthatstupid, fly away anytime the boy and salt
come near. Finally a great truth dawns on the child – you can’tsalt the tail of a bird you haven’t already caught...