Assume there is an isotropic magnetic material in a homogeneous field $\vec{H}$ inside of a long coil. The work done to the material is given by $\delta A = H dM$, where $M$ is the magnetisation of the material.
I'm now supposed to write the entropy of the System as $S = S(T,H)$ and from there get a connection between the magnetisation $M = M(T,H)$ (thermal state equation) and the inner energy $U = U(T,H)$ (caloric state equation)
I did this so far: $dS = (dU - \delta A)\frac{1}{T} = \frac{1}{T} ((\frac{\delta U}{\delta T})_H dT + (\frac{\delta U}{\delta H})_T dH - HdM)$ Because $dS$ is exact it needs to hold that:
$\frac{\delta}{\delta H} [\frac{1}{T} (\frac{\delta U}{\delta T})_H - H (\frac{\delta M}{\delta T})_H] = \frac{\delta}{\delta T} [ \frac{1}{T} (\frac{\delta U}{\delta H})_T - H (\frac{\delta M}{\delta H})_T] $ which in the end yields:
$T^2 (\frac{\delta M}{\delta T})_H = (\frac{\delta U}{\delta H})_T$
If this is incorrect please tell me where I went wrong because now I need to show that when $M = K \frac{H}{T}$, $U = U(T)$. But I get weird results when I plug M into the above equation:
$T^2 (\frac{\delta (K \frac{H}{T})}{\delta T})_H = (\frac{\delta U}{\delta H})_T = T^2 (- \frac{KH}{T^2}) = (\frac{\delta U}{\delta H})_T = -KH$ which doesn't imply that $U$ doesn't depend on H :(
Hope somebody can help me
Greetings!
