This is a crowd pleaser with subtle physics. The upward force on the ring requires that the coil’s B-field have a radial component, and this arises only because the field fringes as shown in the diagram above. The upward launching force on the ring is given by:

$$\textbf{F}=I\textbf{L}\times\textbf{B}_r$$

where $I$ is the induced current in the ring with direction given by Lenz’s Law, $\textbf{L}$ is the current segment length in the direction of the current, and $\textbf{B}_r$ is the radial component of the coil’s magnetic field. Note that there is also an axial component of $\textbf{B}$ (not shown), which exerts a compressive force on the ring but with unseen effect because of the ring’s stiffness. The reason this demonstration works with AC supplied to the coil is due to a phase difference between the supplied AC and the current induced in the ring.