# Experiment: Magnet falling in a copper/aluminum pipe

We know that when we release a magnet in conducting pipe (aluminum, copper, etc..) it will be subjected to a magnetic force induced by the induced current. This magnetic field will be created in a manner opposite to the rate change of flux caused by the falling magnet. This in turn will slow down the magnet as you can see here.

I have seen on this forum message #6, the equation of the terminal velocity reached by the magnet during the fall where $$v$$ is the terminal velocity, $$m$$ is the mass of the magnet, $$g$$ is gravitational acceleration, $$\rho$$ is the resistivity of the tube, $$R$$ is the inner diameter (I am not sure why it is symbolized by R), $$B$$ is the magnetic flux density, $$b$$ is the length of the tube and $$T$$ is its thickness:

$$v = \frac{mg\rho}{4 \pi RB^2bT}$$

My questions are:

1. I have only seen the above equation of Physics Forums, I am not sure how it is derived? If this is difficult to answer, I would like to know if it has a name or if you have seen this equation before?

2. I want to perform this experiment and I would be capable of getting all the above factors except for the magnetic flux $$B$$. Do you have any idea how I can determine B experimentally?

• What would "the" magnetic flux density be? The field will be wildly inhomogenous around the magnet. The easies way to measure the flux would be to use a hall sensor. – Jasper Aug 13 '19 at 12:01
• – Farcher Aug 13 '19 at 12:25
• In the equation from the forum post, it's not clear which value of $B$ to use; the magnetic field of a dipole varies from point to point in space. Another derivation (which is somewhat long & involved) is an exercise in Zangwill's Modern Electrodynamics; the result (up to a proportionality factor) is provided in my answer here. But that derivation depends on several assumptions which may or may not be valid. – Michael Seifert Aug 13 '19 at 13:19
• Has anyone checked the velocity equation for dimensional consistency? I have tried at least 3 times, and can't seem to verify that equation. – David White Aug 13 '19 at 15:58
• @DavidWhite: The units do work out. I'm not sure how accurate it is, though; the equation I posted in the other answer has different dependencies. – Michael Seifert Aug 14 '19 at 12:02

In the first few inches, the magnet will be decelerating as it induces a current in the copper tube. This deceleration will not be constant, as it will be velocity dependent. However, it is safe to assume that the length and time that it takes for the magnet to reach a constant velocity will be the same for both tubes. That length and time will be unknown, but an equation can be set up for the time that it takes for the magnet to fall through each tube. When these two equations are subtracted from each other, everything that remains constant between the tubes, including the distance and time that it takes for the magnet to reach a constant velocity, drops out of the answer. The resulting equation will be: $$t_2 - t_1 = (L_2 - L_1)/v_{constant}$$, where $$v_{constant}$$ is the terminal velocity of the magnet as it falls down the tube.