= Solution
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 is
$$
F=3.333\times10^{-15}Q\,\mathrm N,\qquad \boxed{z=F/k_s=0.2684Q\,\mathrm m.}
$$
Using the persistence-length-corrected <worm-like chain> instead gives \b[$z=1.074Q\,\mathrm m$] 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 $F\sim6.67\times10^{-16}\,\mathrm N$, \b[$z\sim5.37\,\mathrm{cm}$] for part (a)'s model, or \b[$z\sim21.5\,\mathrm{cm}$] 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 $F=5.51\times10^{-17}\,\mathrm N$ 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.
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