= Solution
Applying <Gauss's law for magnetism> to a pillbox and <Ampère's law> to an interface-crossing rectangle gives
$$
\boxed{\hat n\cdot(\mathbf B_2-\mathbf B_1)=0,\qquad
\hat n\times(\mathbf H_2-\mathbf H_1)=\mathbf K_f.}
$$
Here $\hat n$ points from material 1 to 2 and $\mathbf K_f$ is free surface current. With no such current, $B_n$ and $H_t$ are continuous. For linear isotropic media, $B_{t2}/\mu_2=B_{t1}/\mu_1$ and $\mu_2H_{n2}=\mu_1H_{n1}$. The <magnetic fields> need not have equal magnitudes on either side.
With negligible leakage, <Gauss's law for magnetism> makes each cross-section of a <magnetic flux> tube carry the same flux $\mathcal F$. High permeability guides it through the core; a short gap permits neglect of fringing. Flux conservation across these particular geometric sections is an approximation, not a consequence of zero divergence without controlling side leakage.
Let $L$ be the constant-radius arc marked in the figure, and $\mu_r=\mu/\mu_0$. For each straight taper, the <radius> changes linearly from $R$ to $r$, giving the <reluctance of a conical magnetic pole>
$$
\int_{\rm taper}\frac{dx}{\mu\pi a(x)^2}=\frac{s}{\mu\pi Rr}.
$$
Summing <magnetic reluctances> to find the <gap field of a tapered electromagnet>, and using $B_g=\mathcal F/(\pi r^2)$ gives
$$
NI=\mathcal F\left(\frac{L}{\mu\pi R^2}+\frac{2s}{\mu\pi Rr}+\frac{h}{\mu_0\pi r^2}\right),\qquad
\boxed{H_g=\frac{NI}{h+\mu_r^{-1}(Lr^2/R^2+2sr/R)},\quad B_g=\mu_0H_g.}
$$
If $L$ is instead defined as the total magnetic-material <length>, replace the displayed uniform-arc <length> by $L-2s$. The comparison untapered core has uniform <radius> $R$ and material <length> $L+2s$, so
$$
\boxed{\frac{H_g}{H_{g0}}=
\frac{h+(L+2s)/\mu_r}{h+(Lr^2/R^2+2sr/R)/\mu_r}.}
$$
\b[The original PDF specifies $s/L\ll r/R$]; the TeX transcription reverses this inequality. In the printed limit the taper reluctances are small relative to that of the broad arc. The gain is approximately $[h+L/\mu_r]/[h+Lr^2/(\mu_rR^2)]$, approaching $(R/r)^2$ if the gap reluctance is also negligible. If the gap dominates, both fields instead approach $NI/h$ and the gain approaches one. <magnetic saturation> near the small poles, gap fringing, leakage and winding heating limit practical enhancement.
Back to article page
