Tripple well potential with speaker magnet

If a potential has a minimum it gives the stable equilibrium position and the system can oscillate about it. Potentials with more than one minima are much more interesting as it allows hopping between different stable positions and so on.

A speaker magnet gives rise to a cylindrically symmetric magnetic field. What kind of potential will it provide to small ferromagnetic object, which is allowed to move on the circular boundary on its curved surface? Obviously it will be a constant potential and the object can be stuck to any point on this surface. But real interesting potential results if you fix one such object on the magnet and look for the potential on a similar other object.

Procedure

You need a speaker magnet and two ball bearing ball.

1. Stick a ball bearing ball on the inner curved surface of the magnet. Do it at several places and see that it sticks at any point on this surface. This is because of the cylindrical symmetry of the magnet.
2. Let the ball stick at one position and take another similar ball bearing ball. Try to stick it at different positions on the inner curved surface. At how many positions does the ball stick? There are 3 such position.
3. At each such position, shift the ball a little and release. What do you see of the ball?
4. Can you qualitatively find, at which of the 3 positions, is the restoring force (or the force constant) largest. For this position the frequency of oscillation about the equilibrium position should be largest.
5. From your observation, qualitatively draw a plot of potential vs angular position of the second magnet taking the first magnet to be at \(\theta=0\).
6. Remove both the balls. Repeat steps 1-3 with balls placed on the outer curved surface. Write your observations.

Variant: Hold a ball bearing at the center of a speaker magnet

A speaker magnet produces cylindrically symmetric magnetic field. So if you place a ferromagnetic material right at the center, the force on it should be zero. But you may not be able to keep it at the center as it gives an unstable equilibrium position. This experiment explores such situations.

You need a speaker magnet, a ball bearing ball, a bicycle spoke.

1. Keep the speaker magnet in horizontal position and try to put a ball bearing ball at the center. Are you able to put it there? What does the ball do when you release it?
2. Let the ball stick on the inner surface of the magnet. Hold the spoke in a vertical position. Touch one end of the spoke to the ball and carefully shift it towards the center. Can you detach the ball from the magnet?
3. Shift the spoke with attached ball to bring it at the center of the magnet. How does the presence of spoke help in keeping the ball at the center of the speaker magnet?
4. Detach the ball from the spoke. Keep the spoke along the axis of the speaker magnet and try to stick the ball to it. Are you able to stick the ball to the spoke? What is the difference in dragging the ball from the magnet surface to the center with the help of spoke and bringing the spoke first and sticking the ball to it.
5. Hold the speaker magnet in one hand and bring the magnet at the center of the speaker magnet with the help of the spoke. Gradually lift the spoke up. What do you find? Does the ball fall down?
6. In the same situation of step 5, gradually lower down the spoke. What do you find? Does the ball fall down?

Spoke is itself a ferromagnetic material and gets magnetized in the combined field of the speaker magnet and the magnetized ball. The net effect is that the force by the spoke on the ball dominates over the force on it by the magnet as you pull it towards center.

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