Near-contact van der Waals attraction of spheres ($U(h)\simeq -A_{H}a/(12h)$)

For two equal spheres of radius $a$ separated by a surface gap $h\ll a$, a positive Hamaker constant $A_H$ gives the nonretarded Van der Waals force approximation $U=-A_{H}a/(12h)$ and $F=-dU/dh=-A_{H}a/(12h^2)$. The Derjaguin approximation derives the geometry factor from the planar interaction energy per area $W=-A_H/(12\pi h^2)$ and effective radius $a/2$. A sphere and plane instead have effective radius $a$, doubling the force at the same gap. Molecular repulsion, roughness and retardation limit these continuum formulas.