Example: Where is the conservation of angular momentum? (Magnetism)

[MRKN]

Imagine a metal hoop floating in a plane parallel to the Earth's surface, with a permanent magnet suspended above it. The magnet is then dropped straight through the center of the hoop. From Faraday's Law, a current is induced in the hoop as the magnet passes through. There is now angular momentum about the center of the hoop due to the mass of the electrons and their velocities, or current. Whereas before when the magnet is stationary/suspended there is no angular moment as nothing is moving.

Where is the conservation of angular momentum?

 

[Meir Achuz]

There will be angular momentum in the electromagnetic field.
Including that conserves angular momentum.

 

[kcdodd]

its not even that complicated for a neutral wire (imagine dropping the wire instead of the magnet). The protons also feel a force, so the wire itself will move in the opposite direction. If the wire is non-neutral, then you can factor in the angular momentum of the EM field.

 

[MRKN]

Clem- Is it not true that fields do not actually exist? Rather they are a sort of mathematical tool. I don't see how a field can have angular momentum. This seems like a mere mathematical convenience. Could you please refer me somewhere that would explain how fields can have momentum.

 

[Ben Niehoff]

Yes, it is possible to completely eliminate the fields from Maxwell's equations, and rewrite them solely in terms of interactions between charged objects and currents. One can (formally) solve Maxwell's equations for the fields, by means of Jefimenko's equations:

http://en.wikipedia.org/wiki/Jefimenko's_equations

Finally, using the Lorentz force law and Newton's second law, one can eliminate the fields entirely, and obtain expressions for the accelerations of charged particles due to all other charged particles. You get a nasty long expression that is not terribly useful.

The catch is that in this model, all fields must be sourced by charges and currents. So for example, you can't just have a B field going through a loop; you must also include the magnet that produces the B field. Or to simplify things, imagine that you have a second loop, which has a current in it and thus produces a magnetic field.

When you drop Loop 2 through Loop 1, a changing B field induces a current in Loop 1. But note, that current itself produces a B field! This B field, in turn, induces a current in Loop 2, which, if you are careful, you should find is in the opposite direction from the original current in Loop 2.

So effectively, Loop 2 has given up some of its angular momentum to Loop 1; overall, angular momentum is conserved.

In your example, replace Loop 2 by a permanent magnet. Permanent magnets are created by tiny current loops, due to electrons orbiting the nuclei of atoms. When Loop 1 induces the counter-current in the magnet, it has two effects: 1) The magnet, if not perfectly conductive, will start slowly spinning, and 2) Part of the magnet's bound currents will be canceled, and it will decrease its strength.

 

[kcdodd]

I see this assertion a lot, with no explanation for radiation and other such phenomenon. There are cases where you cannot eleminate the idea of fields.

 

[MRKN]

Thanks Ben for the very useful conceptual answer. If you wouldn't mind answering a follow-up question.

Should we constrain the rotation of the permanent magnet, it will be forced to decrease it's magnetic strength proportionally to the speed at which is travels (higher speeds mean higher rates of change in magnetic flux). What would happen if the magnet were to fly past at 1/3 the speed of light?

What will be the limiting factor if the magnet is dropped through a superconductor (still constrained to no rotation)?

What it be possible, with a very long slender rod (very low mass moment of inertia) (possibly cylindrically laminated, as in AC machinery to reduce eddy currents) to have nearly only rotation of the long slender rod and no reduction in magnetic field strength? What now if we bent this long slender rod into a hoop itself s.t. it could not rotate?

 

[Ben Niehoff]

I see this assertion a lot, with no explanation for radiation and other such phenomenon. There are cases where you cannot eleminate the idea of fields.

Jefimenko's equations are predicated on the assumption that no radiation is coming in from infinity. However, radiation can still go out to infinity. So yes, maybe it isn't completely correct to re-absorb E and B by use of the Lorentz force law. It is still possible for the system to radiate momentum (both linear and angular) away to infinity. Even if there is some object at infinity to "catch" the radiation, one can still ask what happens to that momentum in between.

So yes, in the end, one must ascribe momentum to the fields themselves. However, in the case of one current loop traveling at constant velocity through another, there is no radiation.

 

[kcdodd]

I don't agree even in this simple case. In your example, loop 2 will not react to loop 1 in the way you describe. Strictly speaking, both loops will change their current. But the momentum gained by loop 1 is not equal to that lost by loop 2.

(edit: i am mistaken, ExB points radially in the neutral case, but i'll leave it as written) As the current builds up in loop 1, an ExB field will radiate some angular momentum away, so momentum is not even conserved mechanically.

There is no Newtonain "reaction force". And if there was one it would have to travel backwards through time to conserve momentum, or you would have to use a non-local theory, which seems to defeat the purpose of disposing of a field theory, and would probably not even give you the same answers.

 

[Ben Niehoff]

I agree that some momentum will be stored in the fields during the process. The initially-empty loop gains an induced current that goes as 1/r^4, but the counter-induced current in the other loop goes as 1/r^8, so they can't possibly match. I'm not sure if any actual radiation fields get set up or not (that is, fields with a 1/r dependence, such that they radiate away to infinity). It's possible that momentum is stored in the fields and then returned to the loops, if you consider the asymptotic cases. But without trying to calculate, I can't be sure.