Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 235
(a) If B = 4 tesla, at what temperature is 75% of the spins oriented in the +z direction?
(b) Assume that N = 10 23 , T = 1K, and B is increased quasistatically from 1tesla to 10tesla.
What is the magnitude of the energy transfer from the heat bath?
(c) If the system is now thermally isolated at T = 1K and B is quasistatically decreased from
10tesla to 1tesla, what is the final temperature of the system? This process is known as
adiabatic demagnetization. (This problem can be solved without elaborate calculations.)
5.3 Thermodynamics of Magnetism
The fundamental magnetic field is B. However, we can usually control only the part of B due to
currents in wires, and cannot directly control that part of the field due to the magnetic dipoles in
a material. Thus, we define a new field H by
H = 1
µ0
B− M
, (5.21)
V
where M is the magnetization and V is the volume of the system. In this section we use V instead
of N to make contact with standard notation in electromagnetism. Our goal in this section is to
find the magnetic equivalent of the thermodynamic relation dW = −PdV in terms of H, which
we can control, and M, which we can measure. To gain insight into how to do so we consider a
solenoid of length L and n turns per unit length with a magnetic material inside. When a current
I flows in the solenoid, there is an emf E generated in the solenoid wires. The power or rate at
which work is done on the magnetic substance is −EI. By Faraday’s law we know that
E = − dΦ
dt
= −ALndB,
(5.22)
dt
where the cross-sectionalarea of the solenoid is A, the magnetic flux through each turn is Φ = BA,
and there are Ln turns. The work done on the system is
dW = −EIdt = ALnIdB. (5.23)
Ampere’s law can be used to find that the field H within the ideal solenoid is uniform and is given
by
H = nI. (5.24)
Hence, (5.23) becomes
We use (5.21) to express (5.25) as
dW = ALHdB = VHdB. (5.25)
dW = µ0VHdH +µ0HdM. (5.26)
The first term on the right-hand side of (5.26) refers only to the field energy, which would be there
even if there wereno magnetic material inside the solenoid. Thus, for the purpose of understanding
the thermodynamics of the magnetic material, we can neglect the first term and write
dW = µ0HdM. (5.27)