Sphere-plane dispersion force ($F\simeq Aa/(6d^2)$)

For a sphere of radius $a$ at separation $d\ll a$ from a half-space, pairwise integration of the attractive $r^{-6}$ molecular potential gives $F\simeq Aa/(6d^2)$. Equivalently, integrate the planar disjoining pressure magnitude $A/(6\pi h^3)$ over the local gap $h=d+r^2/(2a)$. Across a third medium the effective Hamaker constant can instead correspond to repulsion; state its sign convention explicitly.