PX157 - D1f - a falling magnet in a pipe

TLDR:
- flux at the top decreases, and at the bottom increases
- emf at the top is positive, and at the bottom is negative
- eddy currents are generated in each part: CCW at the top and CW at the bottom
- magnetic fields induced: same polarity as magnet at the top, and opposite polarity at the bottom
- the top part attracts the magnet as it falls into the pipe
- once inside, the top part keeps attracting the magnet while the bottom part repels it, slowing it down
- reaches a terminal velocity once the deceleration equals $g$
- there is also a radial force, but they cancel out
a magnet with a magnetic dipole moment , ${\vec{μ}}_{M}$ , in the $z$ -direction, and a surface current, $I_{M}$ , is in the $ϕ$ -direction
the magnet has a mass, $m$ , and falls due to gravity, $\vec{g} = - g \hat{z}$
magnetic flux:
- at $z = z_{a} :$ the magnetic flux decreases, $\frac{d ϕ_{B}}{d t} (z_{a}) < 0$
- at $z = z_{b} :$ the magnetic flux increases, $\frac{d ϕ_{B}}{d t} (z_{b}) > 0$

from faraday's law:
- at $z = z_{a} :$ an emf, $ϵ$ , generated is counter-clockwise
- at $z = z_{b} :$ and emf, $ϵ$ , generated is clockwise
from ohm's law:
- the material has a finite resistance, $R$
- $ϵ$ drives a circular (eddy) current in the same direction
  - at $z = z_{a} : I = \frac{ϵ}{R}$ counter-clockwise
  - at $z = z_{b} : I = \frac{ϵ}{R}$ clockwise

from ampere's law:
- at $z = z_{a} : {\vec{μ}}_{i} \sim \hat{z}$
- at $z = z_{b} : {\vec{μ}}_{i} \sim - \hat{z}$

\begin{aligned} {\vec{F}}_{i} \sim I_{M} \hat{ϕ} \times {\vec{B}}_{i} & = I_{M} \hat{ϕ} \times (B_{i R} \hat{R} + B_{i z} \hat{z}) \\ = - I_{M} B_{i R} \hat{z} + I_{M} B_{i z} \hat{R} \end{aligned}

{\vec{F}}_{i} = I_{M} | B_{i R} | \hat{z}

the magnet feels an additional force pushing it upwards, acting as a frictional force whose strength is dependent on the resistivity of the material, and is proportional to the magnet's velocity