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## Homework Statement

The phase transition to a superconducting state moves to lower temperatures as the applied magnetic field, H, increases. The magnetic moment, M, for a system of volume V is given by: $$M=-\frac{HV}{4\pi}$$

for ##H<H_C(T)## (superconducting) and $$M=0$$ ##H>H_C(T)## (normal). Changes in the internal energy, U, of such a system can be represented by the expression: $$dU=TdS+HdM$$ a) Show that for this system the constant field and constant magnetization heat capacities are equal: ##C_H=C_M## b) The phase transition between superconducting and normal phases takes place along a path ##H_C(T)##. Find an expression for the slope of the critical field, ##dH_C(T)/dT##, in terms of ##H_C(T)##, ##V## and the entropies of the two phases at the transitions.

## Homework Equations

## The Attempt at a Solution

a) So they give a solution, with a pretty long derivation and I think I am missing something. The way I was planning to do it was like this: $$C_H=\frac{dU}{dT}|_H=\frac{dU}{dT}|_{-4\pi M/V}$$ Assuming the volume is constant, which I assume is the case by "a system of volume V" this is equivalent to $$C_H=\frac{dU}{dT}|_M=C_M$$ I assume something is wrong, otherwise they wouldn't have a much longer derivation, but I am not sure what is wrong. b) I was trying to use the Gibbs free energy, making it equal for normal and superconducting at the boundary between the 2. So I have $$G=U+pV-TS$$ and going to the differential form I get $$dG=dU+pdV+Vdp-TdS-SdT = TdS+HdM-TdS-SdT=HdM-SdT$$ where I assumed that the pressure and volume are constant. However in their solution they get $$dG=-SdT-MdH$$ I assume that $$HdM=MdH$$ by the formula that connects them, but I am not sure how do they get a minus sign there. Can someone help me here?