= Optical trap drag calibration
{title2=$k\langle x\rangle=\zeta u$}
Translate the liquid past a bead held in a stationary <harmonic optical trap>. At steady mean position, take $u$ as the local fluid <velocity> relative to the fixed trap and $x$ positive downstream. Averaging the overdamped <force balance> gives
$$
0=-k\langle x\rangle+\zeta u,\qquad
\langle x\rangle=\frac{\zeta}{k}u,\qquad
k=\frac{\zeta}{d\langle x\rangle/du}.
$$
For the invented <Optical trap Langevin calibration> example, $u=50.0\,\mathrm{\mu m/s}$ predicts $\langle x\rangle=23.562\,\mathrm{nm}$, $V-V_0=0.23562\,\mathrm V$, and $\zeta u=k\langle x\rangle=0.47124\,\mathrm{pN}$. Reversing $u$ reverses the mean shift but leaves the ideal thermal <variance> unchanged. A symmetric speed sweep tests the offset and the linear <force> range; the slope determines $\zeta/k$ if position is calibrated.
Near a wall, $\zeta=6\pi\eta a$ is not the appropriate drag without a boundary correction; record bead height and direction of motion. Stage <velocity> must correspond to the local fluid <velocity> after transients have settled. Escape under increasing flow estimates a maximum holding <force> in that geometry, whereas small-displacement drag calibration estimates the local <spring constant>. They are different quantities. The <atomic force microscopy> similarly needs a calibrated <spring constant> to convert a measured deflection to <force>, but its cantilever is a physical spring rather than this optical field.
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