= Solution
A high-<numerical aperture> objective focuses a laser to a steep intensity gradient. For a transparent particle with higher <refractive index> than its surroundings, refraction transfers photon <momentum> to the particle, giving a <optical gradient force> towards the intensity maximum. A forward scattering <force> pushes along the beam; a stable single-beam <optical trap> requires the restoring axial <optical gradient force> to exceed that push. In the small-particle limit, the same result follows from the induced-dipole <potential energy>, proportional to minus the local squared electric field. Close to the trap centre,
$$
F_x\simeq-kx,\qquad U(x)\simeq\frac12kx^2.
$$
An <optical tweezers> experiment can attach a bead to one end of a <DNA> molecule, hold the other end fixed, and obtain its <force-extension relation> from the bead displacement and calibrated trap stiffness. The measurement must distinguish the restoring <force> from <Brownian motion> and account for heating or optical damage.
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