= Inductance of a toroid with an oblique narrow air gap
{title2=$L\simeq\mu_0N^2A/[\ell/\mu+t\sin\theta]$}
Let a slender <toroid> have core cross-section $A$, mean length $\ell$ and large relative <magnetic permeability> $\mu$. Cut a narrow gap of perpendicular width $t$ whose faces make angle $\theta$ with the local core direction. Continuity of normal <magnetic flux density> gives $B_{g,n}=B\sin\theta$; continuity of tangential <magnetic field intensity> gives $H_{g,t}=B\cos\theta/(\mu\mu_0)$. The gap field is nearly normal to the faces for large $\mu$. Its effective flux area is $A/\sin\theta$, so its leading <magnetic reluctance> is $t\sin\theta/(\mu_0A)$. Adding core reluctance gives the displayed <self-inductance>. At $45^\circ$, $L\simeq\mu_0N^2A/(\ell/\mu+t/\sqrt2)$ and $H_g\simeq NI/(t+\sqrt2\ell/\mu)$. This retains $\mu t/\ell$ while neglecting fringing and corrections of relative order $t/\ell$ or $1/\mu$.
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