= Solution
For two equal spheres, the <Derjaguin approximation> uses $R_{\rm eff}=R/2$. A planar nonretarded <Van der Waals force> interaction has <energy> per area $W(h)=-A_H/(12\pi h^2)$, so
$$
F=2\pi R_{\rm eff}W(h)=-\frac{A_HR}{12h^2}
=-\frac{(15.6\times10^{-20})(10^{-3})}{12(10^{-9})^2}
=\boxed{-1.30\times10^{-5}\ \mathrm N}.
$$
The <force> is \b[attractive, with magnitude $13.0\ \mathrm{\mu N}$], taking separation increase as positive. Using the sphere–plane value of $R_{\rm eff}$ would overestimate it by a factor of two. The gap is much smaller than either radius, as required. Treat the supplied <Hamaker constant> as the effective one for the intervening medium; roughness, solvation and molecular-scale contact can modify a literal $1\ \mathrm{nm}$ experiment.
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