= Solution
Let $\ell=2\pi r-d$ be the mean path length in the core. Neglect leakage and fringing across the narrow air gap, and assume a cross-section small compared with $r$. Continuity of normal <magnetic flux density> then gives the same tangential <magnetic field> magnitude $B$ in the core and gap. <Ampère's circuital law> for the <magnetic field strength> $H$ is
$$
H_{\rm core}\ell+H_{\rm gap}d=nI.
$$
For a linear core, $H_{\rm core}=B/(\mu_0\mu_r)$ and $H_{\rm gap}=B/\mu_0$, giving
$$
\boxed{\mathbf B_{\rm core}\simeq\mathbf B_{\rm gap}\simeq\frac{\mu_0nI}{d+\ell/\mu_r}\,\widehat{\mathbf t}.}
$$
Here $\widehat{\mathbf t}$ follows the winding's <right-hand rule>. The gap has much larger $H$ than the core when $\mu_r\gg1$, although $B$ is approximately continuous. Equivalently, the core and gap <magnetic reluctances> add in series.
On a <magnetic hysteresis> loop, <remanence> is the nonzero $B_r$ remaining at $H=0$ after magnetizing the sample. The <normal magnetic coercivity> $H_{cB}$ is the magnitude of reverse $H$ at which $B=0$ on that branch. The intrinsic <magnetic coercivity> instead refers to zero <magnetization>; these two conventions should not be confused. A comparison of coercive $H$ with remanent $B$ requires a unit conversion or a specified constitutive scale, because their <SI> units differ. A soft <ferromagnet> is understood here to have a narrow loop and small coercive field.
\Image[/past-exam-of-the-electromagnetism-and-optics-course-of-the-university-of-oxford/2016-magnetic-loop-and-reversible-law.png]
{title=Magnetic hysteresis and the reversible tanh approximation}
{description=Original illustrative <magnetic hysteresis> loop with the remanent <magnetic field> and normal coercive <magnetic field strength> marked. The dashed single-valued curve is the reversible approximation used in the calculation, rather than a hysteresis branch.}
Use the stated single-valued <constitutive equation> for the calculation. Inverting it gives $H_{\rm core}=H_m\operatorname{artanh}(B/B_m)$, so
$$
nI=\ell H_m\operatorname{artanh}(B/B_m)+\frac d{\mu_0}B.
$$
Since $\operatorname{artanh}z=z+O(z^3)$, the <small-signal response of a gapped magnetic core> is
$$
\boxed{B\simeq\kappa I,\qquad \kappa=\frac{n}{\ell H_m/B_m+d/\mu_0}=\frac{\mu_0n}{d+\mu_0\ell H_m/B_m}.}
$$
The small-field core <magnetic permeability> is $B_m/H_m$. Strictly, $B=B_m\tanh(H/H_m)$ is a reversible approximation: at $H=0$ it has $B=0$, so it cannot itself model nonzero <remanence>. The displayed loop and this constitutive approximation serve different purposes. Keeping a genuinely remanent branch would generally add a history-dependent offset; it would not imply $B=\kappa I$ through the origin.
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