= Solution
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 $\Phi_2-\Phi_1=-\Phi_{\rm side}$. Neglecting lateral leakage gives
$$
\boxed{\Phi\simeq B(x)A(x)\simeq\text{constant}.}
$$
The 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.
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