= Solution
\b[<Magnetization> and field response.] A magnetic material contains microscopic magnetic moments associated with <Electron> <spin> and orbital motion. The <magnetization> $\mathbf M$ is <magnetic dipole moment> per unit <volume (thermodynamics)>. In SI the macroscopic fields satisfy
$$
\mathbf B=\mu_0(\mathbf H+\mathbf M).
$$
Here $\mathbf H$ separates free-current excitation from the material response. A uniform isotropic linear material has $\mathbf M=\chi_m\mathbf H$, giving
$$
\boxed{\mathbf B=\mu\mathbf H,\qquad\mu_r=1+\chi_m.}
$$
This constant scalar <magnetic permeability> model is useful only within its linear range. Crystals can have a susceptibility <tensor>; strongly magnetic materials depend on field and magnetic history. For a finite specimen the internal $\mathbf H$ also includes its demagnetizing field. In a uniformly magnetized ellipsoid, $H_{\mathrm{int}}=H_{\mathrm{applied}}-N_dM$ along a <principal axis>, so sample shape affects the apparent susceptibility.
\b[Diamagnetic and paramagnetic response.] <Diamagnetism> is a weak negative susceptibility caused by induced orbital response opposing the applied field. It occurs in all matter but can be masked by larger positive contributions. Closed-shell materials such as bismuth or many organic compounds show it clearly. Ordinary <diamagnetism> is often only weakly <temperature> dependent; superconducting flux exclusion is a separate collective phenomenon and is not described by an arbitrarily small negative susceptibility.
<Paramagnetism> arises when available magnetic moments partially align with a field. Thermal disorder prevents complete alignment at weak fields. For independent localized moments of magnitude $m$, the classical weak-field result is
$$
M\simeq\frac{nm^2B}{3k_BT},\qquad
\chi_m\simeq\frac{\mu_0nm^2}{3k_BT},
$$
where the second expression uses $B\simeq\mu_0H$ in a weakly magnetic sample. Thus localized moments obey a <Curie law> in this regime. Quantum <angular momentum> changes the coefficient to the appropriate $J(J+1)$ expression and gives saturation at strong field. Conduction <Electrons> instead exhibit <Pauli paramagnetism>: only states near the <Fermi surface> can change <spin> populations, so a degenerate gas has a nearly temperature-independent response. Its orbital <diamagnetism> must also be included for a full metallic susceptibility.
\b[Collective magnetic order.] <Exchange interactions> can favor parallel alignment, producing <ferromagnetism> below the <Curie temperature>. Iron, cobalt and nickel are familiar examples. A material may contain regions of spontaneous <magnetization>, or <magnetic domains>, while having little net external moment because the domains point in different directions. Applied fields move domain walls and rotate moments; at large fields the <magnetization> approaches saturation. Pinning and <energy> barriers make this process hysteretic. <Remanence> is the <magnetization> left after removing the driving field, and <coercivity> measures the reverse field needed to bring the chosen response back to zero.
The <area> of a closed $B$–$H$ loop is the <energy> dissipated per unit <volume (thermodynamics)> per quasistatic cycle, $\oint H\,dB$, with positive dissipated magnitude. Consequently the <magnetic permeability> is neither one universal constant nor enough information to describe a ferromagnet: differential <magnetic permeability> $dB/dH$, saturation and <hysteresis> all matter. Above the ordering <temperature> the spontaneous moment disappears, and a Curie–Weiss law is often an appropriate approximate paramagnetic description away from the critical region.
In <antiferromagnetism>, ordered opposing sublattices have equal moments and cancel macroscopically in the ideal compensated case. The order disappears above the <Néel temperature>. In <ferrimagnetism>, opposing sublattices have unequal moments and leave a net <magnetization>. Ferrites combine ferrimagnetic response with relatively high electrical resistivity, making them valuable when eddy-current losses would be substantial in a metallic core.
\b[<Electromagnets> and engineering choice.] A coil supplies free-current magnetomotive <force> $NI$. For a simple uniform magnetic path, <Ampère's law> gives $H\ell\simeq NI$ and flux $\Phi\simeq\mu ANI/\ell$. A high-<magnetic permeability> soft magnetic core concentrates flux and reduces the current required for a given field. With an air gap $g$,
$$
NI\simeq\frac{B\ell_c}{\mu_0\mu_r}+\frac{Bg}{\mu_0}.
$$
Even a short gap can dominate the <magnetic reluctance>. The approximate attractive <force> across one uniform gap is $F\simeq B^2A/(2\mu_0)$, obtained from the field <energy>, with fringing neglected.
Soft magnetic materials have small <coercivity> and narrow <hysteresis> loops; they suit relays, <electrical transformer> cores and controllable <electromagnets>. Hard magnetic materials retain their <magnetization> and suit <permanent magnets>. Laminating metallic cores breaks large eddy-current loops; ferrites are useful at higher <frequencies> because of their resistivity. Increasing current cannot increase $B$ indefinitely through the linear formula: core saturation, coil heating and leakage limit performance. A good design selects material for the field amplitude, switching <frequency> and acceptable loss, rather than simply choosing the largest quoted <magnetic permeability>.
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