W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 291

due to an applied magnetic field that causes their angular momentum toprecess around the magnetic
field, theyalsoexperience many small “random” torques due to thermal (heat) fluctuations in their
environment. These torques caused by e.g. collisions between atoms or vibrations in a lattice
constantly more or less randomly reorient the magnetic moments athigh temperatures so that the
system has no net average magnetic dipole moment. A lattice of “spins” at high temperature is
pictured in figure 108.

Bext Bext

a) High temperature b) Low temperature

Figure 108: A lattice of “spins” at high temperature (a) and low temperature (b) is portrayed as
a two dimensional cartoon. The direction of the arrows can be thought of as the directions of the
angular momentum and hence magnetic moment of each atom, in a sideview that reveals their
rough degree of alignment with the field. At high temperature the spins are more or less randomly
aligned with the field, but at low temperature there is less free energy and the spins are much more
likely to be in a lower energy state, partially or completely aligned with the external field.

At low temperatures there is less (free) energy to share among allof the spins – recall that the
equipartition theorem(for example) relates the total kinetic plus potential energy in all of the degrees
of freedom of an atom to its temperature. It is therefore a lot more likely to find the atoms in states
that have “less” magnetic potential energy in the field than those that have more, and atoms have
the least magnetic potential energy when they are in alignment with the field! Consequently, at
low enough temperatures we are likely to find the “permanent” magnetic moments of the atoms or
molecules (if any)aligned with the applied external field!

This alignment causes theexact opposite responseof the material to the field. Since all of the
magnetic moments are lined upwiththe field, and can be much larger than induced magnetic
moments that oppose it that are being createdat the same time, the net field produced by the
“current loops” still cancels on the interior and adds up on the surface, but this time toenhanceor
augmentthe applied field. The total magnetic field inside the material islargerthan the original
external magnetic field. This is portrayed in figure 109.

This kind of response is calledparamagnetism. A paramagnet increases the strength of the
magnetic field inside. Since this (in turn) increases the magneticfluxthrough the material, putting
a paramagnetic material inside a solenoid increases its self-inductance the same way a dielectric
material increases the capacitance of a capacitor. Most solenoidsin electronics use some sort of
paramagnetic material (or ferromagnetic material, read on) to enhance the inductance of their
inductors, getting the same inductance with fewer turns, material, and resistance.

Ferromagnetism and Antiferromagnetism

One canbarelyappreciate paramagnetism classically. Spinning electrons and orbitswith both angu-
lar momentum and a magnetic moment are classically accessible, even though their properties (such
as quantization of the angular momentum) are partly determined byquantum theory. Not so for
the next two kinds of magnetic behavior of materials. They are purely quantum mechanical; one