= Solution
The generic instrument uses a magnetically responsive bead, a controllable external field and optical tracking of bead position or orientation. A field gradient supplies translational <force>; rotating an appropriate anisotropic field or bead moment can supply <torque>. For a permanent dipole,
$$
U=-\mathbf m\cdot\mathbf B,\qquad\mathbf F=\nabla(\mathbf m\cdot\mathbf B),\qquad\boldsymbol\tau=\mathbf m\times\mathbf B.
$$
For an isotropic unsaturated induced moment with effective susceptibility contrast $\Delta\chi$ and bead volume $V$, the <induced magnetic bead force> instead follows from
$$
U=-\frac{V\Delta\chi}{2\mu_0}B^2,\qquad\boxed{\mathbf F=\frac{V\Delta\chi}{2\mu_0}\nabla B^2.}
$$
The factor one-half accounts for inducing the moment. A spatially uniform field alone supplies no translational <force>; an isotropic induced moment parallel to the field supplies no orienting <torque> either. Saturation or magnetic anisotropy requires the corresponding <magnetization> model.
Position tracking needs a length calibration, and <force> calibration needs measured field gradients, a known reference load, or calibrated <thermal fluctuations>. For a generic tensioned tether of extension $\ell$, small transverse displacement $x$ costs energy $Fx^2/(2\ell)$, so $F=k_BT\ell/\langle x^2\rangle$ under equilibrium, small-angle and single-mode assumptions. Attachment geometry, bead rotation, wall drag and finite acquisition bandwidth can affect this calibration. These equations describe generic <magnetic tweezers> mechanics, without assigning a biological response.
TODO: biological single-molecule applications are user-deferred; no biological systems or protocols have been researched or solved.
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