= Solution
The microscopic magnetic response becomes a macroscopic field problem through the <magnetization> $\mathbf M$, the <magnetic dipole moment> per unit <volume>. Define
$$
\mathbf B=\mu_0(\mathbf H+\mathbf M),\qquad
\nabla\cdot\mathbf B=0,\qquad \nabla\times\mathbf H=\mathbf J_{\rm free}
$$
in magnetostatics. Bound currents are represented by $\mathbf J_b=\nabla\times\mathbf M$ and surface current $\mathbf K_b=\mathbf M\times\hat{\mathbf n}$. Normal $B$ is continuous across an interface, while $\hat{\mathbf n}\times(\mathbf H_2-\mathbf H_1)=\mathbf K_{\rm free}$. It is generally incorrect to impose continuity of every component of both fields. A linear isotropic material has $\mathbf M=\chi\mathbf H$ and $\mu_r=1+\chi$, but a ferromagnet usually requires a nonlinear, history-dependent constitutive relation.
Internal field differs from externally applied field. For uniform <magnetization> along a <principal axis> of an ellipsoid, the <demagnetizing field> is $-N_dM$ and $H_{\rm int}=H_{\rm applied}-N_dM$. A linear specimen therefore has apparent susceptibility $\chi/(1+N_d\chi)$ relative to the applied field. Shape is especially important for large susceptibility. A long closed <magnetic circuit> reduces stray fields; a bar magnet still has <demagnetizing fields> despite the absence of magnetic monopoles. Equivalent magnetic-pole descriptions are bookkeeping for <magnetization> <divergence>, not free magnetic charge.<Classical Electromagnetism by Richard Fitzpatrick/Permanent Ferromagnets>[Equations (699)–(706): magnetization, scalar potential and effective pole density]{ref}
In <paramagnetism>, permanent localized moments are preferentially aligned by the field, against thermal disorder. For independent spin-one-half moments with projections $\pm\mu$, the two <Zeeman energies> give
$$
Z_1=2\cosh\left(\frac{\mu B}{k_BT}\right),\qquad
M=n\mu\tanh\left(\frac{\mu B}{k_BT}\right),\qquad
\chi\simeq\frac{\mu_0n\mu^2}{k_BT}>0
$$
in the dilute weak-field limit with $B\simeq\mu_0H$. This is a <Curie law>; the response saturates at $n\mu$ when alignment dominates. More generally, <spin> $J$ gives $\chi=\mu_0n(g\mu_B)^2J(J+1)/(3k_BT)$. Interactions can produce a Curie–Weiss response $C/(T-\Theta)$ over a suitable range. Conduction-electron <Pauli paramagnetism> need not show the localized-moment $1/T$ law.
In <diamagnetism>, the field induces a <magnetic moment> opposing it, so the linear susceptibility is negative. The microscopic cause is the change in orbital electronic motion, and a closed-shell insulating material has no permanent atomic moment to orient. This weak response is often only mildly <temperature> dependent away from other transitions. It also exists as a background in materials with a stronger positive response. Perfect flux expulsion in a superconductor is a distinct collective phenomenon and should not be identified with ordinary weak atomic <diamagnetism>.
In <ferromagnetism>, <exchange interactions> stabilize spontaneous magnetic order below the <Curie temperature>. A specimen can have zero net moment while its <magnetic domains> remain locally magnetized in different directions. Domain-wall motion and moment rotation produce a large response; anisotropy and pinning lead to <remanence>, coercivity and <hysteresis>. Increasing field eventually approaches saturation. Heating above the ordering transition removes spontaneous order, although an applied field still induces a response. Soft magnetic cores favor easy reversal and small cyclic loss; hard permanent-magnet materials favor retained <magnetization> and resistance to reversal. These properties concern collective order and barriers, not merely independent moments all following a weak field.<University Physics Volume 2 by Samuel J. Ling and others>[§12.7, Magnetism in Matter: susceptibility, magnetic domains, Curie temperature and hysteresis]{ref}
A practical distinction requires a controlled field sweep and a separate magnetic readout. A toroidal core with drive winding $N_1$, path <length> $l$ and current $I$ has approximately $H=N_1I/l$. A pickup winding measures $\mathcal E_2=-N_2A\,dB/dt$; integrating after calibration gives $B$, and $M=B/\mu_0-H$. Reversing the sweep reveals a loop whose positive loss <area> per <volume> is $\oint H\,dB$. An air-core/empty-holder reference, known geometry and turn count, offset correction, <temperature> control and slower sweeps test background, heating and rate-dependent eddy-current losses. A single <force> or susceptibility reading cannot establish spontaneous order or <hysteresis>. The original curves below are illustrative responses, not measured material data.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2011/paper-1-magnetic-response.png]
{title=Magnetic response and hysteresis}
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