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Choose a coordinate $z$ measured from the sphere centre towards increasing distance from the block, so an atom in the sphere is at height $L+z$. Summing the atom–block potential energy with the sphere's number density givesFor $R/L\ll 1$, a uniformly valid Taylor series over the sphere isThe odd moments vanish by reflection symmetry. The sphere volume is $V_s=4\pi R^3/3$, and its second moment isConsequentlyThe first finite-radius correction is therefore $-4{\pi }^{2}B{n}^2{R}^5/(15L^5)$. There is no relative $R/L$ correction because the first moment about the centre is zero. The leading term is just the number of sphere atoms times the potential at its centre; the correction increases the attraction because the nearer atoms contribute more strongly than the farther atoms lose.

This is the far-field sphere-plane dispersion energy, with Hamaker constant $A_H={\pi }^{2}B{n}^2$. It must be distinguished from the near-contact sphere-plane dispersion force, whose small parameter is the surface gap divided by the radius. Here $L$ is explicitly the centre-to-plane distance.