A small magnet (weight w) falls under gravity down the center of an infinitely long, vertical, conducting tube of radius a, wall thickness t << R, and conductivity σ. Let the tube be concentric with the z-axis and model the magnet as a point dipole with moment m = mẑ. We can find the terminal velocity of the magnet by balancing its weight against the magnetic drag force associated with ohmic loss in the walls of the tube.
(a) At the moment it passes through z = z₀, show that the magnetic flux produced by m through a ring of radius a at height z' is

(b) When the speed v of the dipole is small, argue that the Faraday EMF induced in the ring is

(c) Show that the current induced in the thin slice of tube which includes the ring is

(d) Compute the magnetic drag force F on m by equating the rate at which the force does work to the power dissipated in the walls of the tube by Joule heating,

(e) Find the terminal velocity of the magnet.

Transcribed Image Text:

ФВ Hom a² 3 2 г where r2 = a² + (zo - z')².