= Solution
With uniform section and negligible <magnetic fringing>, constant <magnetic flux> gives equal <magnetic flux densities> in the core and gap:
$$
\mu_0\mu H_{\rm in}=\mu_0H_{\rm gap},\qquad H_{\rm in}=H_{\rm gap}/\mu.
$$
Here $L$ is the core arc length and $h$ the additional gap length. Applying the <magnetostatic Ampère law> around the complete <magnetic circuit>,
$$
H_{\rm in}L+H_{\rm gap}h=NI,
\qquad
H_{\rm gap}\left(\frac L\mu+h\right)=NI.
$$
Consequently,
$$
\boxed{H_{\rm gap}=\frac{\mu NI}{\mu h+L}.}
$$
This tends to $NI/h$ when the gap dominates the <magnetic reluctance>, $\mu h\gg L$, and reduces to the no-gap result as $h\to0$. The gap <magnetic flux density> is $B_{\rm gap}=\mu_0H_{\rm gap}$.
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