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Magnetic fringing is the spreading of magnetic field lines near an air gap or pole edge, so the gap's effective magnetic flux area exceeds its geometrical pole-face area. A gap much shorter than its transverse dimensions permits a leading uniform-field approximation. High core magnetic permeability does not by itself justify neglecting magnetic fringing for a wide gap.

The starting point is Gauss's law for magnetism, $\nabla \cdot \mathbf{B}=0$: a flux tube carries the same magnetic flux through successive transverse sections. To identify that tube approximately with the physical core, one must also make a negligible-leakage approximation. The large core magnetic permeability makes the core route much less reluctant than competing paths through the surrounding air; the small gap further reduces the incentive for a long external return path. The gentle taper, $s\gg R-r$, lets the field follow the changing section with only small transverse corrections.

For two cuts and the intervening lateral surface, the divergence theorem gives ${\Phi }_2-{\Phi }_1=-{\Phi }_{\mathrm{side}}$. Neglecting lateral leakage givesThe normal magnetic flux density is continuous at each core–air interface by part (a), so this same magnetic flux crosses the gap. To use the gap's geometrical area $\pi r^2$ in the later formulas, the magnetic fringing must also be negligible, normally requiring $h\ll r$. This is the small-gap interpretation of the paper's approximation. Large magnetic permeability alone does not eliminate leakage or magnetic fringing for arbitrary gap geometry.

With uniform section and negligible magnetic fringing, constant magnetic flux gives equal magnetic flux densities in the core and gap:Here $L$ is the core arc length and $h$ the additional gap length. Applying the magnetostatic Ampère law around the complete magnetic circuit,Consequently,This tends to $NI/h$ when the gap dominates the magnetic reluctance, $\mu h\gg L$, and reduces to the no-gap result as $h\to 0$. The gap magnetic flux density is $B_{\mathrm{gap}}={\mu }_0{H}_{\mathrm{gap}}$.

Let the approximately conserved magnetic flux be $\Phi$. ThenThe length $L$ shown in the original diagram is the untapered core arc; the two tapers are separate lengths $s$ each. In the untapered core, flux conservation givesFor one taper use distance $u$ from its narrow end. The straight taper in the diagram has local radiusand magnetic field intensityIts contribution to the line integral in the magnetostatic Ampère law is found explicitly:Adding the gap, the untapered arc and both tapers givesThereforeThis uses a linear radius profile, approximately uniform magnetic flux density over each section, a linear constant magnetic permeability, and negligible leakage and magnetic fringing. For a non-linear taper profile $a(u)$ the honest generalization isThus an arbitrary smooth taper with the same end radii and length need not give the printed formula; the straight-sided geometry supplies the additional information.

Set $\alpha =r/R<1$. Comparing with the untapered expression in part (d), using the same $N,I,\mu ,h,L$ and large radius $R$,The stated limit $s/L\ll \alpha$ makes $2s\alpha \ll L{\alpha }^2$, soThe 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.