Gap field of a tapered electromagnet ($H_g=NI/[h+{\mu }_r^{-1}(L{r}^2/R^2+2sr/R)]$)

A uniform-radius magnetic arc of length $L$ and radius $R$, two linear tapers of length $s$ to radius $r$, and an unfringed vacuum gap of length $h$ have the displayed gap field. Here ${\mu }_r=\mu /{\mu }_0$ and $B_g={\mu }_0{H}_g$. Tapering concentrates flux, but at fixed winding current the gap-dominated limit is $H_g\simeq NI/h$ regardless of pole area. Saturation and leakage prevent unrestricted field enhancement.