= Solution
For the generic instrument, <magnetic tweezers> combine a magnetically responsive bead, a controllable <magnetic field gradient>, a tether attached to a fixed support, and optical tracking of bead coordinates. For a fixed <magnetic dipole moment>, the <force> is $\nabla(\mathbf m\cdot\mathbf B)$; an induced linear isotropic moment instead gives $\mathbf F=V\Delta\chi\nabla B^2/(2\mu_0)$. A uniform <magnetic field> can orient an anisotropic bead, but a gradient is needed for translation. Magnet height sets the load, while field rotation can impose <torque> if bead orientation is coupled to the field. Tracking uses lateral images and a calibrated axial image/defocus response; a fixed reference removes stage drift. A sufficiently slowly varying gradient approximates a <force clamp> over the bead's displacement. TODO: \b[user-deferred biomolecular implementation]; DNA-specific attachment, preparation and response are not treated.
For <transverse-fluctuation calibration of a tensioned tether>, use an ideal tether of vertical extension $h$ whose endpoint is displaced by small $x,y$. Keeping its <length> fixed to quadratic order lowers its vertical coordinate by $(x^2+y^2)/(2h)$, increasing load potential by $F_z(x^2+y^2)/(2h)$. The <equipartition theorem> therefore gives
$$
\boxed{F_z=\frac{k_BTh}{\langle\delta x^2\rangle}
=\frac{k_BTh}{\langle\delta y^2\rangle}
=\frac{2k_BTh}{\langle\delta x^2+\delta y^2\rangle}.}
$$
Here fluctuations are about the mean. The height is the effective pendulum <length> of the measured coordinate, not automatically the <polymer contour length>. The displayed $h$ formula uses the point-endpoint or fixed-bead-orientation idealization; freely rotating large beads can require a bead-radius correction to that lever arm. The <finite-exposure correction to fluctuation force calibration> restores variance lost through exposure averaging; ignoring it would overestimate <force>; localization noise increases it and would underestimate <force>. Drift, wall coupling, finite observation time and magnetic lateral stiffness must be checked. Vertical fluctuations measure longitudinal stiffness and do not obey the same geometrical formula.
For the <stretch and twist coefficients of an elastic filament>, take $s$ as reference contour <length> and $\epsilon$ as local axial <strain>. The three positive quadratic costs represent bending of the tangent, stretching of the material and excess twist about its relaxed orientation. With the paper's coefficient convention,
$$
K_b=k_BTA,\qquad S=k_BTB,\qquad K_t=k_BTC.
$$
$K_b,K_t$ are bending and torsional rigidities with units of <energy> times <length>; $S$ is an axial <force> modulus. Thus $A,C$ have units of <length> and $B$ inverse <length>. Positive quadratic forms describe a stable relaxed filament; odd terms vanish when expansion is about its zero-stress, zero-torque reference configuration.
At weak and intermediate <tension>, changing thermal bending conformations can extend the filament without appreciably stretching the backbone. Axial <strain> is $\epsilon\simeq F_z/S$, so material stretching must be included when this is no longer negligible. For a long strongly aligned but nearly inextensible three-dimensional <worm-like chain>, the thermal extension deficit is approximately $\tfrac12\sqrt{k_BT/(AF_z)}$; stretch adds approximately $F_z/S$ to fractional extension. Twist matters whenever <torque> or imposed end rotation is present, at any <tension>. A freely swivelling, <torque-free> tether need not store excess twist. There is no universal force-only threshold for switching on torsional <energy>.
The <circular bending energy of an elastic filament> follows because curvature is $1/R$ over contour <length> $2\pi R$. Integrating the bending term gives
$$
\boxed{E_{\rm loop}=\frac{k_BTA}{2R^2}(2\pi R)=\frac{\pi k_BTA}{R}.}
$$
This is bending <energy> alone: imposed <tension>, stretching and twist would add costs. The dimensionless bending action is $(A/2)\int|\partial_s\mathbf t|^2ds$. Its tangent propagator in three dimensions obeys <rotational diffusion> on the unit sphere, $\partial_sP=(2A)^{-1}\nabla_{S^2}^2P$. The vector <spherical harmonic> has angular <eigenvalue> $-2$, giving
$$
\boxed{\langle\mathbf t(s)\cdot\mathbf t(0)\rangle=e^{-s/A}.}
$$
Hence $A$ is the <persistence length> in this three-dimensional convention. A strictly planar filament instead has tangent-correlation <length> $2A$; one cannot infer the factor merely by setting a loop's <energy> equal to $k_BT$.
TODO: \b[user-deferred biomolecular applications]. Both enzyme-driven unwinding experiments, their experimental sketches, the requested numerical <force> and plectoneme <radius>, <energy> loss per turn and enzyme-specific dissipation mechanisms remain unsolved. None of their biomolecular parameters has been used in a calculation.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2013/paper-1-generic-magnetic-calibration.png]
{title=Original generic magnetic-tether instrument and its transverse fluctuation calibration}
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