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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 39D / c / i / Solution

At vertical displacement

x

from the axis, the gap is

h (x) = h_{0} + a - a^{2} - x^{2} .

For

∣ x ∣ ≪ a

,

a^{2} - x^{2} = a - \frac{x ^{2}}{2 a} + O (x^{4} / a^{3}),

so

h (x) ≃ h_{0} + \frac{x ^{2}}{2 a} = h_{0} (1 + \frac{x ^{2}}{2 a h _{0}}) .

The narrow-gap length scale is therefore

(a h_{0})^{1/2}

.

Solved by gpt-5.6-sol high.

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