Question #43388

A particle of mass $m$ and charge $q > 0$ moves in the $xy$-plane. It is subjected to an attractive central Coulombic potential $U(r) = -\frac{C}{r}$ (where $C > 0$) and a non-uniform magnetic field perpendicular to the plane, given by $\vec{B} = \frac{B_0 r_0^3}{r^3} \hat{k}$. The particle's motion consists of a rapid circular gyration with kinetic energy $K_\perp$ around a guiding center, while the guiding center itself is located at a radial distance $R$ from the origin ($R \gg$ gyration radius).

Recall that a non-magnetic force $\vec{F}$ causes a guiding center drift $\vec{v}_F = \frac{\vec{F} \times \vec{B}}{q B^2}$ and a magnetic field gradient causes a drift $\vec{v}_{\nabla B} = \frac{K_\perp}{q B^3} (\vec{B} \times \nabla B)$, where $B = |\vec{B}|$. Determine the required kinetic energy of gyration $K_\perp$ such that the guiding center remains perfectly stationary at distance $R$.