= Solution
<Magnetization> $\mathbf M$ is <magnetic dipole moment> per unit volume. In SI units,
$$
\mathbf B=\mu_0(\mathbf H+\mathbf M),\qquad
\mathbf M=\chi\mathbf H,\qquad \mu=\mu_0(1+\chi)
$$
in a linear isotropic response. The internal field matters: for a uniformly magnetized <ellipsoid> it is $\mathbf H_{\rm int}=\mathbf H_{\rm applied}-\mathbf N_d\mathbf M$. Sample shape therefore affects an apparent <magnetic susceptibility>.
\b[<Diamagnetism>] opposes the applied field, giving small negative <magnetic susceptibility> from the induced orbital response. It is often only weakly <temperature> dependent. \b[<Paramagnetism>] aligns pre-existing moments and gives positive <magnetic susceptibility>; independent localized moments obey approximately $\chi=C/T$ in weak fields before saturating. <Pauli paramagnetism> of conduction electrons is a different, weakly temperature-dependent mechanism.
\b[<Ferromagnetism>] involves collective exchange-driven alignment and spontaneous <magnetization> below the <Curie temperature>. A specimen can nevertheless have nearly zero total moment because <magnetic domains> point in different directions. Field-driven domain-wall motion and <rotation> change the net <magnetization>. <Antiferromagnets> have compensating ordered sublattices below a <Néel temperature>; <ferrimagnets> have oppositely ordered but unequal sublattice moments and retain a net moment. Ferrites are useful examples of electrically resistive ferrimagnetic core materials.
Ferromagnetic response is nonlinear and history dependent. A <hysteresis> loop displays <remanent magnetization>, <magnetic coercivity> and saturation. A single constant <magnetic permeability> cannot describe the entire loop: differential <magnetic permeability> describes its local slope. The quasistatic <energy> loss per unit volume and cycle is the loop <area> $\oint H\,dB$. <Soft magnetic materials> have low <magnetic coercivity> and usually high useful <magnetic permeability>; <hard magnetic materials> retain <magnetization> and suit permanent magnets.
\b[An <electromagnet> normally uses a soft, high-permeability core] to guide flux generated by a coil. For a uniform-section core with length $\ell_c$ and air gap $g$, neglecting leakage and fringing,
$$
NI\simeq H_c\ell_c+\frac{Bg}{\mu_0},\qquad
\boxed{B\simeq\frac{\mu_0NI}{g+\ell_c/\mu_r}.}
$$
An air gap can dominate the <magnetic reluctance> even when the core <magnetic permeability> is large. Core saturation limits the improvement with current, while winding resistance causes heating. For alternating operation, laminations suppress eddy-current loops in metallic cores; high-resistivity ferrites also reduce those losses. Low <hysteresis> loss is important in repeatedly cycled cores, whereas high <magnetic coercivity> is valuable in a permanent magnet. Useful design therefore depends on the full magnetic response, electrical losses and operating <frequency>, rather than the largest quoted <magnetic permeability> alone.
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