# Work performed by a magnetic field

Here is a little thought experiment related to magnetism and a perplexing question regarding its physics. Suppose we have a long cylinder of transparent plastic, and we press fit and then cement a circular magnet in one end of the cylinder with its north pole oriented into the cylinder. We also have a permanent bar magnet with a square profile the diagonal of which is just small enough for it to slide freely into the cylinder.

We orient the cylinder perpendicular to the earth's surface with the cylindrical magnet at the bottom. We then drop the bar magnet into the top of the cylinder with its north pole pointing downward toward the north pole of the circular magnet at the other end of the tube. What happens is that gravity allows the bar magnet to fall until the upward repulsive forces exactly balance the force of gravity.

This is well known. It is also well-known that we could replace the bottom magnet with an electromagnet. When we energize the electromagnet and drop in the bar magnet, the same phenomenon will be observed. However, in this case, as soon as we remove the current from the bottom magnet, the top magnet will fall.

OK, so here's the confusing part: With the electromagnet, it is easy to calculate the energy used to produce the repelling field because there is a simple relationship between the current flow through the coil and the magnetic force produced:

F=CAni/l

where C is a proportionality constant, A is the cross-sectional area of the plunger, n is the number of turns in the solenoid, I is the current through the solenoid wire, and l is the length of the solenoid. So for a given F we can derive i, and then easily compute the energy in joules required to create that current. Let's say it is X joules.

My question is: in the alternative case of two permanent magnets where does the energy required to produce X joules come from? If we were to leave the two magnets like this with the top magnet floating, will the energy, whatever it is, eventually become depleted so that the bar magnet will closer closer and closer toward the cylindrical magnet?

Staff Emeritus
No work is required to hold an object at rest.

• sophiecentaur
sophiecentaur
Gold Member
2020 Award
No work is required to hold an object at rest.
The repelling magnets situation is only the same energetically as a mass being supported on a spring. The arrangements will both take up a position of minimum Potential Energy.

@ Vanadium 50, There may be no work while it is at rest because it is not moving. But how about the deceleration? It is easy to compute the work required to stop the bar magnet from falling. So in the case of the permanent magnet, where does the energy required to produce THAT work come from?

sophiecentaur
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