Physics 4311: Thermal Physics – HW Solution 10

Problem 1: Entropy in a magnet (10 points)

use Maxwell relation originating from $\mathit{𝑑𝐹}=-\mathit{𝑆𝑑𝑇}-\mathit{𝑚𝑑𝐵}$

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)

$$\mathit{𝑑𝑆}={\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝐵}}\right)}_T\mathit{𝑑𝐵}+{\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝑇}}\right)}_B\mathit{𝑑𝑇}$$

$${\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝑇}}\right)}_m={\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝐵}}\right)}_T{\left(\frac{\mathit{\partial 𝐵}}{\mathit{\partial 𝑇}}\right)}_m+{\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝑇}}\right)}_B$$

$$c_m-c_B=T{\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝑇}}\right)}_m-T{\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝑇}}\right)}_B=T{\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝐵}}\right)}_T{\left(\frac{\mathit{\partial 𝐵}}{\mathit{\partial 𝑇}}\right)}_m$$

use Maxwell relation originating from $\mathit{𝑑𝐹}=-\mathit{𝑆𝑑𝑇}-\mathit{𝑚𝑑𝐵}$

$${\left(\frac{\mathit{\partial 𝑆}}{\mathit{\partial 𝐵}}\right)}_T={\left(\frac{\mathit{\partial 𝑚}}{\mathit{\partial 𝑇}}\right)}_B$$

$$c_m-c_B=T{\left(\frac{\mathit{\partial 𝑚}}{\mathit{\partial 𝑇}}\right)}_B{\left(\frac{\mathit{\partial 𝐵}}{\mathit{\partial 𝑇}}\right)}_m$$

b)

$$m=\frac{\mathit{𝑎𝐵}}{T}{\left(\frac{\mathit{\partial 𝑚}}{\mathit{\partial 𝑇}}\right)}_B=-\frac{\mathit{𝑎𝐵}}{T^2}$$

$$B=\frac{\mathit{𝑇𝑚}}{a}{\left(\frac{\mathit{\partial 𝐵}}{\mathit{\partial 𝑇}}\right)}_m=\frac{m}{a}=\frac{B}{T}$$

$$c_m-c_B=-\frac{a{B}^2}{T^2}$$