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## 24 February 2011

### Induced EMF caused by a falling magnet, and a qualitative demonstration

 Helmholtz coils of the type useful for demonstrating the answer to today's question.
I've seen the multiple-choice question from at least two different sources:  a bar magnet is dropped through a circular wire coil from above.  The north end of the magnet points down.  What will a graph of induced EMF (or current) in the coil vs. time look like?

This question requires use of Lenz's Law.  I teach an approach to Lenz's law described here.  Let's consider first the time while the magnet approaches the wire loop, and then separately the time when the magnet recedes toward the ground.

As the magnet approaches, the magnetic field is down toward the ground because the B field due to a bar magnet points out of the north end.  The magnetic flux is increasing because the bar magnet approaches the coil, increasing the magnetic field at the location of the coil.  Becuase flux increases, I flip my right thumb opposite the magnetic field, and curl my fingers -- the current in the coil is counter-clockwise when viewed from above.

After the magnet passes through the coil and is receding, the magnetic field still points down toward the ground!  This is because the B field due to a bar magnet points into the south end.  The magnetic flux is now decreasing because the bar magnet recedes from the coil, producing an ever-smaller magnetic field at the location of the coil.  Becuase the flux decreases, I keep my right thumb pointing in the direction of the magnetic field, curl my fingers, and find the current in the coil is the other way -- clockwise as viewed from above.

I asked my AP classes this question as a "check your neighbor" exercise today in class.  At first, when working alone, most of the class (~2/3) didn't recognize that the current switches direction when the magnet passes through the coil.  Upon discussion, I found that the key misconception was the direction of the magnetic field when the magnet receded.  Either they thought that the magnetic field must have a different direction near the other pole, or they thought that since the field was decreasing, the field itself must switch directions.  Good arguments among the class changed some views, such that after discussion about 2/3 of the class had the correct answer.

Setting up the demonstration:

Long time blog readers know that I don't think it good enough to explain something like this conceptually and/or mathematically without doing the experiment.  Nature is the ultimate arbiter of arguments.  So I set up the problem.

I used a 10-cm diameter wire coil with a gazillion loops.  I've on other occasions used a bigger diameter set of coils scavenged from a 20 year old broken q/m device.  You'll get the best results if you use a coil with enormous numbers of loops, like one designed to produce strong, uniform magnetic fields -- a Helmholtz coil, like those shown at the top of the post.

I hooked a Vernier voltage probe to my labpro, and set the labpro to take 1000 data points per second for 2 seconds.  A student clicked "collect;"  When data collection began, I dropped a small (~5 cm length) bar magnet through the coil.
A bit of zooming produced the graph shown above.  As expected, the voltage switched polarity (i.e. the current switched directions) as the magnet passed through the coil.

Though it's not brutally obvious, the right-hand hump is not as wide on the graph as the left-hand hump.  Why not?  Because the magnet speeds up in free fall, and so recedes from the coil faster than it approached the coil.  I dropped the magnet from about shoulder height.  The ~0.10 s duration of the voltage spikes are substantially less than the ~0.5 s time I calculate for the magnet to fall to the ground, as it should be.

Any more follow up thoughts?  This demonstration can produce a wealth of interesting calculations and experiments suitable for an end-of-year independent laboratory investigation.

1. Awesome post.

2. do u have a computer model on this ?
i currently only have this http://weelookang.blogspot.com/2011/01/ejs-open-source-magnet-falling-through.html i am trying to find the equations on solenoid and a longer magnet.

3. great explaination!

4. the area of graph under curve to right of point where v=0 should be equal to left one

5. the area of graph under curve to right of point where v=0 should be equal to left one

6. Do somebody has a mathematical equation that descripes the above graph...if u do send me on habby221@gmail.com.

7. Should the voltage of the curve when magnet is leaving the coil be higher? As induced voltage=change in flux/time. The magnet is falling at a faster velocity, hence the time for flux change is lesser, meaning that the magnitude of induced voltage is larger when the magnet is leaving coil.