Page 395 - Electrical Properties of Materials

Thermodynamical treatment 377

Let us now review the thermodynamical equations describing the phase
transitions. There is the first law of thermodynamics: E is the internal energy, W the
work, S the entropy, P the pres-
dE =dQ –d W
sure, V the volume, and Q the
= T dS – P dV. (14.2) heat.
Then there is the Gibbs function (which we shall also refer to as the Gibbs free
energy) defined by

G = E + PV – TS. (14.3)
An infinitesimal change in the Gibbs function gives

dG =dE + P dV + V dP – T dS – S dT, (14.4)
which, using eqn (14.2) reduces to

dG = V dP – S dT. (14.5)

Thus, for an isothermal, isobaric process,
dG =0,

that is, the Gibbs function does not change while the phase transition takes
place.
In the case of the normal-to-superconducting phase transition the variations
of pressure and volume are small and play negligible roles, and so we may just Fig. 14.8
as well forget them but, of course, we shall have to include the work due to The magnetization of magnetic
material in a toroid (for working out
magnetization.
the magnetic energy).
In order to derive a relationship between work and magnetization let us
investigate the simple physical arrangement shown in Fig. 14.8. You know from U is the voltage and I the current,
studying electricity that work done on a system in a time d t is and the negative sign comes from
the accepted convention of ther-
d W =–UI d t. (14.6) modynamics that the work done on
a system is negative.
Further, using Faraday’s law, we have

d B
U = NA . (14.7)
d t
From Ampère’s law,
HL = NI. (14.8)

We then get A is the cross-section of the toroid,
N the number of turns, and L the
d B mean circumference of the toroid.
d W =–NA I d t =–NI A d B =–HL A d B =–VH d B. (14.9)
d t
According to eqn (11.3),

B = μ 0 (H + M).
Therefore,