Physics 4311: Thermal Physics – Homework 10

Problem 1: Entropy in a magnet (8 points)

The magnetic susceptibility of some magnetic solid is given by the Curie-Weiss law, $\chi =A∕(T-T_W)$ where $A$ and the Weiss temperature $T_W$ are constants. How much does the entropy of this solid change if the magnetic field $H$ is increased from 0 to $H_0$ at fixed temperature $T$?

Problem 2: Heat capacities of a magnet (30 points)

This problem is about the heat capacities of a magnetic material. It compares the heat capacity at constant field, $C_B$ with heat capacity at constant magnetization, $C_m$.

a)

Show that for any magnetic material, the two heat capacities are related via

$$C_m-C_B=T{\left(\frac{\mathit{\partial 𝑚}}{\mathit{\partial 𝑇}}\right)}_B{\left(\frac{\mathit{\partial 𝐵}}{\mathit{\partial 𝑇}}\right)}_m .$$

Hint: Start from the total differential of the entropy as a function of $B$ and $T$ and take a derivative w.r.t. $T$ at fixed $m$. Use a Maxwell relation to simplify the resulting expression.

b)

Show that for a paramagnet that fulfills the Curie law $m=\mathit{𝑎𝐵}∕T$, the derivatives in the above relation can be computed, giving $C_m-C_B=-a{B}^2∕T^2$.