Diamagnetic levitation in an inverse-height field ($z_0^3=-\chi (B_0{r}_0{)}^2/({\mu }_0\rho g)$)

A small weakly susceptible sphere has induced magnetic energy $-\chi V{B}^2/(2{\mu }_0)$ and vertical force $\chi VB{B}^{\prime }/{\mu }_0$. In field magnitude $B=B_0{r}_0/z$, balancing gravity gives the title height for $\chi <0$. The vertical force varies as $z^{-3}$, giving a restoring derivative at that equilibrium. Three-dimensional trapping cannot be established from an axial field magnitude alone; lateral field geometry and Maxwell constraints also matter.