 Radial-path approximation for a cracked toroid

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 $B(r)\simeq {\mu }_0{\mu }_{r}NI/[2\pi r+({\mu }_r-1)d]$; integrating across the section gives the displayed mutual inductance. Even $d\ll a$ can strongly reduce inductance when ${\mu }_{r}d/a$ 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 $\hat{\mathbf{\varphi }}/(r+q)$ everywhere in a current-free core is not curl-free; transverse corrections are required. Narrow-gap circuit approximations and their limitations are discussed in ^[ref].