= Solution
A <magnetic field> favors alignment of the <magnetic moments> of a <paramagnet>, reducing the number of substantially occupied magnetic configurations at fixed <temperature>. Its <magnetic entropy> therefore decreases as the field magnitude increases, explaining the negative sign of $(\partial S/\partial B)_T=-\alpha B/T^2$.
Choose the magnetic <Helmholtz free energy> convention $dF=-S\,dT-m\,dB$. Equality of its <mixed partial derivatives> gives the magnetic <Maxwell relation>
$$
\left(\frac{\partial m}{\partial T}\right)_B=\left(\frac{\partial S}{\partial B}\right)_T=-\frac{\alpha B}{T^2}.
$$
The <magnetic moment inferred from an entropy derivative>, integrated at fixed $B$, is
$$
\boxed{m(T,B)=\frac{\alpha B}{T}+m_0(B).}
$$
The given <entropy> derivative alone cannot determine the temperature-independent function $m_0(B)$. With the usual independent-<paramagnet> condition $m\to0$ as $T\to\infty$ at fixed $B$, $m_0(B)=0$, giving <Curie's law>, $m=\alpha B/T$. The sketch uses that additional condition. The inverse-$T$ expression applies in the weak-field regime; its divergence cannot describe the actual low-<temperature> saturation of a finite set of <magnetic moments>.
\Image[/past-exam-of-the-thermal-physics-course-of-the-university-of-oxford/2016-paramagnetic-response.png]
{title=Inverse-temperature moment at fixed magnetic field}
{description=Original <Curie's law> sketch under the condition of zero moment at infinite <temperature>. The low-temperature saturation regime is outside this approximation.}
For reversible quasistatic thermal isolation, $dS=0$. With positive field-dependent <heat capacity> $C_B=T(\partial S/\partial T)_B$,
$$
0=\frac{C_B}{T}dT-\frac{\alpha B}{T^2}dB,\qquad
\boxed{dT=\frac{\alpha B}{C_BT}\,dB.}
$$
Reducing a positive $B$ therefore lowers $T$: this is <adiabatic demagnetization>. Equivalently, reducing $B^2$ cools for either field sign. Reduced alignment would raise <magnetic entropy> at fixed $T$; cooling compensates so total <entropy> remains constant. Insulation without reversibility would allow additional <entropy production> and need not follow this adiabat exactly.
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