Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 71 / 5 / c

An optical tweezers force requires the change in optical momentum. Define the optical momentum efficiency $Q$ by $F=QP/c$, with all ray geometry and medium-index factors absorbed in $Q$ and $c$ the vacuum light speed. The specified relative refractive index does not determine $Q$ without bead size and the beam's convergence, waist, and ray directions. A unique exact force is therefore not supplied by these data.

Retaining the entropic stiffness from part (a), the parameterized answer isUsing the persistence-length-corrected worm-like chain instead gives $z=1.074Q\mathrm{m}$ in its small-force regime.

For a definite scale estimate, take the weak-contrast optical momentum change to be $Q\sim m_r-1=0.2$. This gives $F\sim 6.67\times 10^{-16}\mathrm{N}$, $z\sim 5.37\mathrm{cm}$ for part (a)'s model, or $z\sim 21.5\mathrm{cm}$ for the corrected chain. These are estimates with an explicitly chosen ray-momentum efficiency, not a universal formula for a transparent trapped sphere.

Another illustrative optical model makes the uncertainty visible: normal-incidence Fresnel reflectance is $\mathcal{R}=[(m_r-1)/(m_r+1){]}^2=1/121$. An effective reflector with $Q=2\mathcal{R}$ in the simple $P/c$ momentum convention gives $F=5.51\times 10^{-17}\mathrm{N}$ and extensions $4.44\mathrm{mm}$ or $17.7\mathrm{mm}$, respectively. A refracting bead in a focused beam is not just that reflector; entry and exit rays also transfer momentum. Neither estimate eliminates the missing optical geometry.

The extension must be compared with the actual contour length, $L=100\mathrm{m}$. For the illustrative $Q=0.2$ force and the persistence-length-corrected chain, $z/L\simeq 2.15\times 10^{-3}$ and $F{b}_K/(k_{B}T)\simeq 6.44\times 10^{-3}$. Both are small, so the small-force Langevin function expansion and entropic-extension approximation are consistent. Although the mean extension greatly exceeds the unforced millimetre-scale coil size, it remains far below the maximum contour length; that does not violate the model.

The material axial elongation at this force would be only $F/(EA/L)\simeq 5.0\times 10^{-5}\mathrm{m}$, illustrating that most of the predicted coil extension is orientation and straightening rather than stretching the backbone. Thus the order-of-magnitude entropic response is internally reasonable within the ideal-chain assumptions, but its numerical optical-force prediction is not unique. Solvent-dependent excluded volume, the unusually large stated contour length, actual trap geometry, and surface interactions all matter to an experimental interpretation.