= Solution
\b[Inductance and interface conditions.] For a linear coil, <self-inductance> is the ratio $L=\Lambda/I$ of its own <magnetic flux linkage> $\Lambda=N\Phi$ to current. Its induced <electromotive force> is $-L\,dI/dt$ for fixed $L$. Across an interface with unit normal $\mathbf n$ from medium 1 to 2,
$$
\boxed{\mathbf n\cdot(\mathbf B_2-\mathbf B_1)=0,\qquad
\mathbf n\times(\mathbf H_2-\mathbf H_1)=\mathbf K_f.}
$$
These <electromagnetic boundary conditions> follow from <Gauss's law for magnetism> and <Ampère's law>. Here $\mathbf K_f$ is <free surface> current. At the iron–air faces there is no <free surface> current, so normal <magnetic flux density> and tangential <magnetic field intensity> are continuous. In each isotropic linear medium, $\mathbf B=\mu_0\mu_r\mathbf H$.
\b[Complete core.] A slender <toroid> has approximately uniform $H$ along mean path length $\ell$. <Ampère's law> gives $H\ell=NI$, so $B=\mu\mu_0NI/\ell$. The flux per turn is $\Phi=BA$, yielding the <inductance of a toroidal winding>,
$$
\boxed{L_0=\frac{N\Phi}{I}=\frac{\mu\mu_0N^2A}{\ell}.}
$$
This uses constant relative <magnetic permeability> $\mu$ and neglects leakage, fringing and saturation.
\b[Why the gap lowers inductance.] The gap replaces an easy magnetic path in iron with air. Maintaining the same flux now requires a much larger <magnetic field intensity> in part of the path and therefore more ampere-turns. Equivalently, <magnetic reluctance> increases, so the flux linkage for a fixed current and hence <self-inductance> decrease. A narrow gap can matter greatly because its permeability is smaller by the factor $\mu$.
\b[Oblique-gap calculation.] In the original diagram $t$ is the perpendicular separation of the cut faces. Let $\mathbf s$ point along the core, and let $\mathbf n,\mathbf q$ be normal and tangential to the gap faces. With the face at angle $\theta$ to the core direction, $\mathbf s=\sin\theta\,\mathbf n+\cos\theta\,\mathbf q$. Write the core flux density as $B\mathbf s$. The boundary conditions give
$$
B_{g,n}=B\sin\theta,\qquad
H_{g,q}=\frac{B\cos\theta}{\mu\mu_0},
$$
so
$$
\mathbf H_g=\frac B{\mu_0}
\left(\sin\theta\,\mathbf n+\frac{\cos\theta}{\mu}\mathbf q\right).
$$
The gap area normal to its nearly normal field is $A/\sin\theta$, not $A$. Its leading reluctance is consequently $t\sin\theta/(\mu_0A)$. Adding the core reluctance gives the <inductance of a toroid with an oblique narrow air gap>,
$$
NI\simeq\frac B{\mu_0}
\left(\frac\ell\mu+t\sin\theta\right),
\qquad L\simeq\frac{\mu_0N^2A}{\ell/\mu+t\sin\theta}.
$$
Thus at $\theta=45^\circ$,
$$
\boxed{L\simeq\frac{\mu\mu_0N^2A}{\ell+\mu t/\sqrt2}.}
$$
The approximation retains the possibly order-one ratio $\mu t/\ell$ while neglecting $t/\ell$, fringing and the small tangential gap contribution. More precisely, a local uniform-field path along the core, with missing iron length $t/\sin\theta$ subtracted, gives denominator $\ell/\mu+t\sin\theta(1-1/\mu)$. Its correction is beyond the stipulated large-$\mu$, narrow-gap accuracy; simply neglecting $\mu t$ would not be justified.
\b[Field in the slot.] The component expression above gives
$$
H_g=\frac B{\mu_0}\sqrt{\sin^2\theta+\frac{\cos^2\theta}{\mu^2}}.
$$
For the $45^\circ$ gap and $\mu\gg1$,
$$
\boxed{H_g\simeq\frac{B}{\sqrt2\mu_0}
\simeq\frac{NI}{t+\sqrt2\ell/\mu}.}
$$
Its units are $\mathrm{A\,m^{-1}}$. The field is almost perpendicular to the cut faces, rather than following the iron direction straight across the cut. Its small angle away from the face normal satisfies $\tan\alpha=1/\mu$ at $45^\circ$; the sense reverses if the current reverses.
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