= Radial-path approximation for a cracked toroid
{title2=$M_d\simeq\frac{\mu_0\mu_rN_1N_2h}{2\pi}\ln\frac{b+q}{a+q},\quad q=\frac{(\mu_r-1)d}{2\pi}$}
In this <magnetic circuit> approximation, each circular path has air-gap <length> $d$, core <length> $2\pi r-d$, and the same normal <magnetic flux density> in air and core. <Ampère's law> then gives $B(r)\simeq\mu_0\mu_rNI/[2\pi r+(\mu_r-1)d]$; integrating across the section gives the displayed <mutual inductance>. Even $d\ll a$ can strongly reduce inductance when $\mu_rd/a$ is appreciable. This is a path estimate, not an exact local field solution in the slit geometry. It neglects <magnetic fringing> and redistribution between paths. In particular, a literal field proportional to $\widehat{\boldsymbol\phi}/(r+q)$ everywhere in a current-free core is not curl-free; transverse corrections are required. Narrow-gap circuit approximations and their limitations are discussed in <Electromagnetic Fields and Energy by Hermann A. Haus and James R. Melcher/Magnetic Circuits>[§9.7, Example 9.7.1; paragraph following the example]{ref}.
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