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We have to find proportionality of the terminal velocity with the factors of the system: enter image description here

Plot: a small dipole(mass $m$) with dipole moment $\mu$ is dropped in a long solenoid (radius $r$, Resistance $R\: \mathrm{\Omega}$) along it's axis. The gravity is acting downwards.

I found it to be $$v\propto \dfrac{mR}{\mu^2r}$$

Here is how:

The force in downward direction $F_{down}$ must be $mg$ and in upward direction $F_{up}$ must be magnetic force. $$F_{up}\propto \dfrac{\mu I_{induced}}{r}$$

$$I_{induced}\propto\dfrac{ \text{electric flux change}}R \propto\dfrac{\mu \pi r^2 v}R$$

Flux change depends directly on the velocity, so added $v$ in the numerator. But the expression is not correct. The proportionality on the $r$ radius is not correct. I think there is a problem in the induced current factor.

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