A very cold torsion pendulum

A thin aluminum ring hangs vertically from a torsion spring. A torsion spring when twisted exerts a restoring torque given by $\tau=- \kappa \theta$ where $\theta$ is the angle of twist. Suppose that the ring undergoes small torsional oscillations while it is being cooled down to the point where it becomes superconducting. The period of torsional oscillations of the superconducting ring is $T_{0}$. This period changes after applying an external horizontal magnetic field of induction $B$ parallel to the plane of the ring corresponding to $\theta=0$ (the position of equilibrium). Show that for the case of a weak magnetic field $B$, the new period of oscillations is $T=T_{0}-\Delta T \quad \textrm{with} \quad \Delta T= C\frac{a^{4} T_{0}^3 B^{2}}{J L}.$ Here, $a$ is the radius of the cold ring, $J$ is the moment of inertia with respect to the vertical axis ($J=\frac{1}{2}m a^{2}$), $L$ is the ring's self inductance and $C$ is a numerical coefficient. Determine the coefficient $C$.

Details and assumptions

Hint: $(1+x)^{\alpha}\approx 1+\alpha x \quad \textrm{for} \quad x\ll 1.$

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