478
8 Electric and Magnetic Phenomena
Maxwell and Maxwell-like relations arising from the combined 1st and 2nd
Laws expression (8.4.6) in conjunction with the Gibbs energy G(T , P , B 0 ) ≡
U(S, V , M) + P V − T S − MB 0 .
7. Show that the entropy, S spin , of a system of N distinguishable spin1
2 particles
in an applied magnetic induction field of strength B 0 and given by Eq. (8.5.14)
may also be written as
S spin (N, T , B 0 ) =
1
2 Nk B
ln
4(Nμ N g I ) 2
(Nμ N g I ) 2 − 4M 2 (T , B 0 )
−
4M(T , B 0 )
Nμ N g I
tanh
−1
2M(T , B 0 )
Nμ N g I
.
What consequence does this result imply for C M ?
8. Use Eq. (8.4.13b) to establish that the heat capacity at constant magnetic
moment, C M , vanishes for a system of N distinguishable spin1
2 particles.
9. Starting from expression (8.4.1) for the magnetic field contribution to the
internal energy, show that its contribution to the combined 1st and 2nd Laws
expression for dU is given by dU field = B 0 · dM if the magnetizable
thermodynamic system is of fixed volume V and has fixed permeability μ.
What condition(s) is(are) thus placed upon the magnetic susceptibility of the
material?
10. Starting from the combined 1st and 2nd laws expression dU = T dS + B dM,
and by considering both the internal energy U and the entropy S to be functions
of temperature T and magnetic moment M, show that the heat transferred
between a macroscopic system and its surroundings is given by
δQ rev = C B (T ) dT + T
∂M
∂T
B
dB.
11. If the equation of state for a magnetizable sample is given by M = χ(T )B
with χ(T ) known as the magnetic susceptibility, show that for an adiabatic
demagnetization process (in which the magnetic induction is decreased by an
amount dB), the temperature change, dT , is given by
dT = −
BT
C B
dχ
dT
dB.
Many paramagnetic crystals have a susceptibility given by the Curie law,
χ(T ) =
Nμ 2
m
3k B T
,
in which N is the number of paramagnetic molecules or ions in the crystal, and
μ m represents the magnetic moment of an individual molecule or ion. Deter-
8 Electric and Magnetic Phenomena
Maxwell and Maxwell-like relations arising from the combined 1st and 2nd
Laws expression (8.4.6) in conjunction with the Gibbs energy G(T , P , B 0 ) ≡
U(S, V , M) + P V − T S − MB 0 .
7. Show that the entropy, S spin , of a system of N distinguishable spin1
2 particles
in an applied magnetic induction field of strength B 0 and given by Eq. (8.5.14)
may also be written as
S spin (N, T , B 0 ) =
1
2 Nk B
ln
4(Nμ N g I ) 2
(Nμ N g I ) 2 − 4M 2 (T , B 0 )
−
4M(T , B 0 )
Nμ N g I
tanh
−1
2M(T , B 0 )
Nμ N g I
.
What consequence does this result imply for C M ?
8. Use Eq. (8.4.13b) to establish that the heat capacity at constant magnetic
moment, C M , vanishes for a system of N distinguishable spin1
2 particles.
9. Starting from expression (8.4.1) for the magnetic field contribution to the
internal energy, show that its contribution to the combined 1st and 2nd Laws
expression for dU is given by dU field = B 0 · dM if the magnetizable
thermodynamic system is of fixed volume V and has fixed permeability μ.
What condition(s) is(are) thus placed upon the magnetic susceptibility of the
material?
10. Starting from the combined 1st and 2nd laws expression dU = T dS + B dM,
and by considering both the internal energy U and the entropy S to be functions
of temperature T and magnetic moment M, show that the heat transferred
between a macroscopic system and its surroundings is given by
δQ rev = C B (T ) dT + T
∂M
∂T
B
dB.
11. If the equation of state for a magnetizable sample is given by M = χ(T )B
with χ(T ) known as the magnetic susceptibility, show that for an adiabatic
demagnetization process (in which the magnetic induction is decreased by an
amount dB), the temperature change, dT , is given by
dT = −
BT
C B
dχ
dT
dB.
Many paramagnetic crystals have a susceptibility given by the Curie law,
χ(T ) =
Nμ 2
m
3k B T
,
in which N is the number of paramagnetic molecules or ions in the crystal, and
μ m represents the magnetic moment of an individual molecule or ion. Deter-