= Solution
The tension is supplied by <Stokes drag> on the free bead. With bead <radius> $b=0.5\,\mu\mathrm m$, the stipulated isolated-sphere approximation gives
$$
\boxed{f=6\pi\eta bU=2.83\,\mathrm{pN}.}
$$
The trap also balances drag on the trapped bead, but that bead's drag is not an additional <force> transmitted through the chain. The <polymer contour length> is $L=Na=6\,\mu\mathrm m$. Use a room-<temperature> estimate $T\simeq300\,\mathrm K$, consistent with the water-viscosity approximation imported from part (a); the expressions below retain the dependence on $T$.
For a <theta solvent>, <Gaussian chain> <elasticity> gives $F_{\rm el}=3k_BTR^2/(2Na^2)$ and hence
$$
\boxed{R_\theta=\frac{fNa^2}{3k_BT}
=0.819\left(\frac{300\,\mathrm K}{T}\right)\,\mu\mathrm m.}
$$
This is the mean extension along the <force>, rather than the unforced <end-to-end distance of a polymer chain> $a\sqrt N=60\,\mathrm{nm}$. At $300\,\mathrm K$, $fa/(k_BT)=0.410$; the finite-link <freely jointed chain> result $R=L[\coth(fa/k_BT)-k_BT/(fa)]$ is $0.810\,\mu\mathrm m$, close to the Gaussian estimate.
The <good-solvent force-extension scaling> changes the <elasticity>. In the <tensile blob> regime, a blob has size $\xi_t=k_BT/f$, contains $g\sim(\xi_t/a)^{5/3}$ segments, and the aligned string of $N/g$ blobs has <length>
$$
\boxed{R_{\rm good}\sim\frac Ng\xi_t
=Na\left(\frac{fa}{k_BT}\right)^{2/3}
\simeq3.31\left(\frac{300\,\mathrm K}{T}\right)^{2/3}\,\mu\mathrm m.}
$$
Only the scaling and order of magnitude of the good-solvent value are determined: solvent-dependent swelling amplitudes are not supplied. The nonlinear <force> law is the essential correction to extrapolating a weak-force coil <spring>.<Excluded Volume Effects and Stretched Polymer Chains by Philip Alan Pincus>[abstract and section I, p. 386: strongly elongated good-solvent chain and restoring force proportional to extension to the power 3/2]{ref}
To expose the unrealistic extrapolation, the unforced good-solvent size estimate is $R_0\sim aN^{3/5}=151\,\mathrm{nm}$. Treating this coil as a linear <spring> with <spring constant> $3k_BT/R_0^2$ predicts $R\simeq5.17\,\mu\mathrm m$ at $300\,\mathrm K$, although its required condition $fR_0\ll k_BT$ fails by a factor of about $103$. Dropping the order-one factor three, as in the rough estimate $k\sim k_BT/R_0^2$, even gives $R\sim15.5\,\mu\mathrm m>L$. \b[An extension beyond the <polymer contour length> is an artefact of extending the coil's <linear response> outside its range.] The <tensile blob> estimate avoids that violation but here $\xi_t\simeq1.46\,\mathrm{nm}$ and $g\sim4.4$, so it is only marginally within a many-segment-blob regime. <finite extensibility of a polymer chain>, chain drag and flow disturbance by both large beads further limit quantitative realism. The specified neglect of chain drag is a model assumption, not evidence that it is physically negligible.
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