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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.