A P P E N D I X Â
F R E E E N E R G Y O F A M A G N E T I C B O D Y
SUPPOSE that a body of volume V is uniformly magnetized by an applied field Ha, so that its magnetic moment is M. T h e field is now increased to Ha + dHa and produces a change of magnetic moment dM. T h e total energy which must be supplied (e.g. by a battery driving a coil which generates the magnetic field) to make these increments in field and magnetic moment at constant temperature i s t
dWtot = V. Ü(\ìïÇ2á) + ì0ÇáÜÌ.
T h e first term on the right-hand side represents the work done in increasing the strength of the applied field over the volume occupied by the body; this work must be done to increase the field whether the body is present or not. T h e second term, ì0ÇáÜÌ, is the energy which must be supplied to increase the magnetic moment of the body. L e t us call this dWM9
dWM = ì0ÇáÜÌ.
Comparison of this with the work done by an external pressure p in producing an infinitesimal change in volume dV of a body,
dWv = -pdVf
shows that the expression for the work to magnetize the body h a s a similar form, if w e suppose that ìïÇ á corresponds top and Ì to — V.t
N o w the G i b b s free energy for a body in the absence of a magnetic field is
G=U-TS + pVy
t See, for example , A. B. Pippard , Classical Thermodynamics y Cambridg e Universit y Press , 1957, p. 26.
X Th e signs ar e differen t becaus e energ y mus t be given to a bod y to increas e its magnetization , wherea s wor k is don e by a bod y whe n it increase s in volum e agains t an externa l pressure .
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236 INTRODUCTION TO SUPERCONDUCTIVITY
where U and S are its internal energy and entropy. As we might expect from the above considerations, the effect of magnetization is to add a term —ì0ÇáÌ analogous to the term +pV:
G= U-TS + ñí -ìïÇáÌ.
Small changes in the conditions will produce a change in G given by dG = dU- TdS - SdT + pdV + Vdp - ì0ÇáÜÌ - ì0ÌÜÇá. If the applied field Ha and the magnetic m o m e n t Ì are changed while the temperature and pressure are kept constant (dT = dp = 0), we have
dG = dU- TdS + pdV - ì0ÇáÜÌ - ì0ÌÜÇá. But for a magnetic body under the same conditions of constant Ô a n d / ) ,
dU = TdS - pdV + ì0ÇáÜÌ.
work done on body
Therefore dG = —ì0ÌÜÇá, and the change in free energy of a body when it is magnetized to a magnetic m o m e n t Ì by a field of strength Ha
is
G(Ha)-G(0) = -MoJMdHa. 0
