Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2020 / Paper 1 / Section B / 10 / iii

Under uniform geometric scaling, write ${\mathbf{r}}_i^{\prime }=\lambda {\mathbf{r}}_i$. Each volume element contributes a factor ${\lambda }^3$, while the inverse-sixth-power separation contributes ${\lambda }^{-6}$. The double integral therefore scales as ${\lambda }^3{\lambda }^3{\lambda }^{-6}=1$, andThis assumes unchanged material Hamaker constant, pairwise-additive nonretarded interactions and the same intervening medium. It is a simultaneous scaling of both body sizes and their separation, not merely moving fixed bodies apart.

For nonretarded attraction between planar half-spaces, the interaction free energy per area is $W(h)=-A_H/(12\pi h^2)$. The Derjaguin approximation gives $F=2\pi R_{\mathrm{eff}}W(h)$, with $R_{\mathrm{eff}}=R_1{R}_2/(R_1+R_2)=a/2$ for equal spheres. ThusThe force is $13.0\mu \mathrm{N}$, attractive. Scaling both radii and the gap by two gives $F^{\prime }=F/2$, or $6.50\mu \mathrm{N}$ attractive. This agrees with the invariant energy $V\simeq -A_{H}a/(12h)$: differentiating with respect to a gap that doubles makes the force halve. The approximation requires $h\ll a$ and neglects roughness and retardation.