Inductive oscillation of a sliding rod on inclined rails (${\omega }_B^2=B^2{l}^2{\cos }^2\theta /(mL)$)

A vertical field linking a conducting loop on an incline contributes flux $Bl\cos \theta s$. For zero resistance and zero initial current, flux conservation gives $LI=Bl\cos \theta s$. The uphill magnetic force component then makes the downhill coordinate oscillate about $s_e=mgL\sin \theta /(B^2{l}^2{\cos }^2\theta )$. Release from $s=0$ first stops at $2s_e$, not at the force-free point $s_e$. The energy is stored in the inductor, not dissipated.

For positive loop resistance, coupled mechanical and circuit equations relax toward constant sliding speed on sufficiently long rails. The steady current balances the downhill gravitational force, and motional electromotive force balances the resistive voltage. The undamped zero-resistance limit has no relaxation, so its late-time limit is singular.