Demagnetization with a linear-heat-capacity load ($T_f=T_i-(N\theta /M)\Delta s_{\mathrm{mag}}/k_B$)

Let $N$ independent spin-$J$ moments and a load with heat capacity $C_L=M{k}_{B}T/\theta$ remain in equilibrium during reversible isolated demagnetization. Then $S_L(T)=S_L(0)+M{k}_{B}T/\theta$. Fully removing the field raises each spin entropy by its initial magnetic entropy deficit. Total entropy conservation gives $N\Delta s_{\mathrm{mag}}=M{k}_B(T_i-T_f)/\theta$, yielding the title result. The formula assumes positive final temperature, validity of the load law throughout the interval, negligible additional entropy contributions and an attainable equilibrated low-field endpoint. Interactions require an actual many-body entropy instead of the independent-spin expression.