= Far-field sphere-plane dispersion energy
{title2=$U(L)=-2A_HR^3(1+6R^2/(5L^2)+\cdots)/(9L^3)$}
For pairwise-additive nonretarded molecular attraction and equal solid <number densities> $n$, let $A_H=\pi^2Bn^2$ and let $L>R$ be the distance from a sphere centre to a planar half-space. Integrating the atom–half-space <potential energy> gives
$$
U(L)=-\frac{A_H}{6}\left[\frac{2RL}{L^2-R^2}+\ln\frac{L-R}{L+R}\right].
$$
For $L\gg R$, expansion gives the displayed expression. Equivalently expand $(L+z)^{-3}$ over the sphere: its odd moments vanish and $\langle z^2\rangle=R^2/5$. This yields the first relative correction $6R^2/(5L^2)$. In the opposite near-contact limit, the gap $h=L-R\ll R$ gives $U\simeq-A_HR/(6h)$ and the <sphere-plane dispersion force> magnitude $A_HR/(6h^2)$. The two limits use different distances and must not be confused.
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