Uniformly magnetized ring with a narrow gap ($H_g=lM/(l+a),H_r=-aM/(l+a)$)

A rod of magnetic path length $l$ and tangential magnetization $M$ closed by a gap of width $a$ has constant flux in the one-dimensional magnetic circuit approximation. Equal cross sections and negligible leakage give $H_g=H_r+M$, while absence of free current gives $l{H}_r+a{H}_g=0$. Their solution is the displayed pair. Gap fringing or an unspecified cross-section change requires a more detailed field model.