= Solution
Set $\alpha=r/R<1$. Comparing with the untapered expression in part (d), using the same $N,I,\mu,h,L$ and large radius $R$,
$$
\frac{H_{\rm gap,taper}}{H_{\rm gap,untaper}}
=\frac{\mu h+L}{\mu h+L\alpha^2+2s\alpha}.
$$
The stated limit $s/L\ll\alpha$ makes $2s\alpha\ll L\alpha^2$, so
$$
\boxed{\frac{H_{\rm gap,taper}}{H_{\rm gap,untaper}}\simeq
\frac{\mu h+L}{\mu h+L\alpha^2}>1.}
$$
The taper concentrates <magnetic flux> into a smaller pole-face area, increasing the gap <magnetic field intensity> and <magnetic flux density> for the same winding <electric current>. If $\mu h\ll L\alpha^2$, the enhancement approaches $1/\alpha^2=(R/r)^2$. If instead $\mu h\gg L$, the gap dominates the <magnetic reluctance> and the enhancement approaches unity: both fields approach $NI/h$.
The benefit is therefore strongest when the untapered core's contribution is significant, while the taper lengths contribute little. The same-core-length convention is the comparison defined by parts (d) and (e); if one instead fixes the full iron path length, the untapered comparator has length $L+2s$, a difference negligible in the stated limit. A smaller gap area can increase the field even though the total <magnetic flux> need not increase. <Magnetic saturation> and <magnetic fringing>, excluded from this model, limit the ideal concentration.
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