= Turning radius of an initially stationary magnetic ring
For negligible natural radius and release from rest at $r_1$, $J=qBr_1^2/2$. The effective radial potential is a sum of $r^{-2}$ and $r^2$ terms, whose two turning radii have squared product $J^2/(m^2\Omega^2)$. Hence
$$
r_{\min}=\frac{|qB|r_1}{\sqrt{q^2B^2+16\pi^2km}},\qquad
\dot\phi_{\min}=\frac{8\pi^2k}{qB},\qquad\dot r_{\min}=0.
$$
These formulas assume $qB\ne0$. At exactly zero field the ideal ring collapses to zero radius instead of making a regular positive-radius turning point.
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