= Solution
The extension must be compared with the actual contour length, \b[$L=100\,\mathrm m$]. For the illustrative $Q=0.2$ <force> and the persistence-length-corrected chain, $z/L\simeq2.15\times10^{-3}$ and $Fb_K/(k_BT)\simeq6.44\times10^{-3}$. 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 $F/(EA/L)\simeq5.0\times10^{-5}\,\mathrm m$, illustrating that most of the predicted coil extension is orientation and straightening rather than stretching the backbone. Thus \b[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.
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