Back when I was playing with a strong magnet dropping through a coil of wire I wondered how much energy I could extract from the dropped magnet and if I could do anything with it. The coil I was using was at least 15 cm in diameter, which means that I wasn’t capturing all of the flux lines from the magnet — the field of a dipole drops off as 1/r^3, so a smaller diameter would be much better and the slowing of the magnet could be noticeable, as we’ve seen before with someone dropping a magnet down a copper tube.

Since I’m a physicist, I wanted to quantify this. I didn’t have a copper tube handy, but I do have a roll of aluminum foil which is on a roll with an inner diameter of about 3.8 cm (1.5 in), which is a reasonably tight fit for my strong magnet. I set up my slow-motion camera and my ipod in stopwatch mode to double-check the timing (yes, it was shooting at a rate of 210 frames per second)

I exported the video to individual frames to make it easier to analyze, and counted frames. The free drop takes about 0.25 seconds, give or take (it’s hard to tell exactly which frame represents release) and I estimate the distance as being about 32 cm (a foot-long roll = 30 cm, with the start just above and stop just below). The drop through the aluminum foil roll takes about 0.38 seconds. The freefall drop is easy to analyze: v = gt, and to double-check for g, just rearrange the familiar kinematics equation and solve. The drop time implies a speed of about 2.45 m/s at the exit. For g we get 10.2 m/s^2, so my little experiment seems good to 10% or better.

For the drop through the tube, we don’t know exactly what’s going on. There’s a damping force that varies with speed and eventually we would expect the magnet to reach terminal velocity. To get an estimate, though, let’s first assume it’s a uniformly lower acceleration. That would give us a value of 4.4 m/s^2 for the acceleration and an exit speed of 1.67 m/s. If we assume it hits terminal speed immediately then the speed would be 0.84 m/s. The truth is somewhere in the middle. There are probably several ways I could test this further, but the ones I can think of either require dropping the magnet from a distance above the tube, and it’s a tight fit, so it probably means lots of trials before I got lucky and got the magnet to drop in, or using a longer tube. I know aluminum foil comes in different lengths, but I only have the one. Since I want an idea of the energy extracted, let’s use the worst case value of 1.67 m/s.

I found the mass of the magnet using a small electronic scale and a plastic cup to keep the magnet away from the metal pan (where it might also be attracted to the interior or the case and mess up the measurement) and subtracted the mass of the cup. 60 grams.

Which means the magnet lost about 0.1 Joules of kinetic energy in the foil, in less than 0.38 seconds, or an average power of just over a quarter of a Watt, in that worst-case scenario. The best-case is 50% higher. And this is using aluminum — copper will give is a better result. Recall that Faraday’s law is
\(V = -frac{dphi}{dt}\)
Copper’s resistivity is about 2/3 of aluminum’s, so a given potential will drive about 50% more current and boost the resistive force owing to the larger field from the additional current. In other words, we can expect copper to be more efficient at converting the mechanical energy to electrical. It will more closely approximate the terminal-speed-quickly scenario, and it should have a smaller terminal speed.

What I want to do in the near future is wind a coil on one of these cardboard tubes and see if I can light up a little light bulb.