428
8 Electric and Magnetic Phenomena
in which the magnetic susceptibility χ (m) is a second rank tensor. We shall, however,
limit ourselves to the linear response regime, for which χ (m) (T , ρ, B 0 ) becomes a
scalar multiple, χ (m) (T , ρ), of the isotropic second rank tensor δ (whose Cartesian
components are Kronecker deltas δ ij , with i, j = x, y, z), so that the macroscopic
magnetic moment M is parallel to B 0 , i.e.,
M = χ
(m) (T , ρ)B 0 .
(8.4.4b)
Note that the magnetic susceptibility may, in principle, still be a function of both
the temperature, T , and mass density, ρ, of the thermodynamic system. Within the
linear response regime the permittivity μ is given in terms of μ 0 and χ (m) by
μ = μ 0 (1 + χ
(m) ).
(8.4.5)
The combined 1st and 2nd Laws expression, Eq. (2.3.12a) is extended to include
the energy change (8.4.3) induced in the system by the constant external magnetic
field, to become
dU = dU + dU field = T dS − P dV + B 0 · dM
(8.4.6)
for a magnetizable thermodynamic system. In the following, we shall focus our
discussion on the thermodynamics of a magnetically isotropic medium, so that M
lies along the direction defined by B 0 , thereby allowing us to replace M everywhere
by M and B 0 everywhere by B 0 .
As the magnetic moment of the macroscopic sample depends upon both temperature, T , and (for an isotropic medium) the magnitude, B 0 , of the external magnetic
flux into which it has been placed, the total differential dM of the macroscopic
magnetic moment may be expressed as
dM =
∂M
∂T
B 0
dT +
∂M
∂B 0
T
dB 0 .
(8.4.7)
The response of a macroscopic collection of magnetic dipoles to a change in
the magnitude of the applied magnetic flux, B 0 , under isothermal conditions is
known as the isothermal magnetic susceptibility, and is denoted by χ (m) (T ). From
expression (8.4.7), we see that χ (m) (T ) is thus given by
χ
(m) (T ) =
∂M
∂B 0
T
.
(8.4.8)
As pointed out by Pippard [6], the thermodynamic results traditionally obtained
for the internal energy U as a function of pressure P and volume V , as well as the
expressions for the constant pressure and constant-volume heat capacities and their
difference, as given in Eq. (2.5.1c) can be transformed into equivalent expressions
8 Electric and Magnetic Phenomena
in which the magnetic susceptibility χ (m) is a second rank tensor. We shall, however,
limit ourselves to the linear response regime, for which χ (m) (T , ρ, B 0 ) becomes a
scalar multiple, χ (m) (T , ρ), of the isotropic second rank tensor δ (whose Cartesian
components are Kronecker deltas δ ij , with i, j = x, y, z), so that the macroscopic
magnetic moment M is parallel to B 0 , i.e.,
M = χ
(m) (T , ρ)B 0 .
(8.4.4b)
Note that the magnetic susceptibility may, in principle, still be a function of both
the temperature, T , and mass density, ρ, of the thermodynamic system. Within the
linear response regime the permittivity μ is given in terms of μ 0 and χ (m) by
μ = μ 0 (1 + χ
(m) ).
(8.4.5)
The combined 1st and 2nd Laws expression, Eq. (2.3.12a) is extended to include
the energy change (8.4.3) induced in the system by the constant external magnetic
field, to become
dU = dU + dU field = T dS − P dV + B 0 · dM
(8.4.6)
for a magnetizable thermodynamic system. In the following, we shall focus our
discussion on the thermodynamics of a magnetically isotropic medium, so that M
lies along the direction defined by B 0 , thereby allowing us to replace M everywhere
by M and B 0 everywhere by B 0 .
As the magnetic moment of the macroscopic sample depends upon both temperature, T , and (for an isotropic medium) the magnitude, B 0 , of the external magnetic
flux into which it has been placed, the total differential dM of the macroscopic
magnetic moment may be expressed as
dM =
∂M
∂T
B 0
dT +
∂M
∂B 0
T
dB 0 .
(8.4.7)
The response of a macroscopic collection of magnetic dipoles to a change in
the magnitude of the applied magnetic flux, B 0 , under isothermal conditions is
known as the isothermal magnetic susceptibility, and is denoted by χ (m) (T ). From
expression (8.4.7), we see that χ (m) (T ) is thus given by
χ
(m) (T ) =
∂M
∂B 0
T
.
(8.4.8)
As pointed out by Pippard [6], the thermodynamic results traditionally obtained
for the internal energy U as a function of pressure P and volume V , as well as the
expressions for the constant pressure and constant-volume heat capacities and their
difference, as given in Eq. (2.5.1c) can be transformed into equivalent expressions