= Lossless limit of magnetic loop braking
{title2=$\mathcal L I+Kz=\text{constant}$}
For an external loop flux $Kz$, positive current orientation and upward <Lorentz force> $KI$, including <self-inductance> gives $\mathcal L\dot I+RI=-K\dot z$ and $m\ddot z=-mg+KI$. At zero resistance total linked flux is conserved. If initially $I=0$ at $z=z_0$, then $I=-K(z-z_0)/\mathcal L$ and the mechanical motion is an undamped oscillator about $z_0-mg\mathcal L/K^2$. The finite-resistance terminal speed tends to zero as $R\to0$, but neglecting <self-inductance> is a singular approximation in that limit. A lossless loop does not dispose of <energy> through <Joule heating>.
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