Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 344 / 3 / f / Solution

For a fixed center height $h$, the radius decreases according to the preceding current enhancement. ThusWithin the stipulated logarithmic approximation, $c(y)=1+y/2+O(y^2)$, so $1/c(y)=1-y/2+O(y^2)$. Expanding uniformly over $0\le R\le R_0\ll h$ and integrating givesTherefore the leading correction to the isolated-droplet lifetime isThe negative correction reflects faster evaporation into the nearby reservoir. This leading coefficient also agrees with the small-$R/h$ expansion of the exact sphere-plane capacitance; the stipulated logarithmic approximation need not match higher orders.