= Demagnetization with a linear-heat-capacity load
{title2=$T_f=T_i-(N\theta/M)\Delta s_{\rm mag}/k_B$}
Let $N$ independent spin-$J$ moments and a load with <heat capacity> $C_L=Mk_BT/\theta$ remain in equilibrium during reversible isolated demagnetization. Then $S_L(T)=S_L(0)+Mk_BT/\theta$. Fully removing the field raises each spin <entropy> by its initial <magnetic entropy deficit>. Total <entropy> conservation gives $N\Delta s_{\rm mag}=Mk_B(T_i-T_f)/\theta$, 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.
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