How Does Canonical Momentum Relate to Angular Momentum Conservation?

[Bob not Alice]

Hi folks,

There are a number of statements out there on the web to the effect that 'canonical momentum is simply “the quantity that is conserved” in electromagnetic interactions, while the kinetic momentum is just the product of mass and velocity'. Fair enough. There must also be a presumption that once the electromagnetic interactions have become so weak as to be negligible kinetic momentum is once more the conserved quantity.

At any point in this post if I'm making silly statements then please feel free to, gently, point them out.

Advanced sums are all fine and good (I did them a very long time ago at Uni) but there's nothing quite like a thought experiment...

Imagine two electromagnets, one of which is nice and powerful and switched on. At some large distance from it is a smaller one, not currently switched on and which is not aligned with the magnetic field lines emanating from the first one but positioned so that the field experienced is essentially uniform. Send a very brief pulse of current through the smaller one and that will initiate a rotational movement in it - the same forces are at work as those which start a compass needle swinging towards north/south. That rotation should continue even though the smaller electromagnet is once more switched off. If it doesn't then please educate me!

Give the system time to settle down and one would expect to see a change in the motion of the larger electromagnet caused by the brief field from the smaller one. Angular momentum is conserved as the induced "electromagnetic interactions have become so weak as to be negligible".

Action at a distance isn't possible so this equalisation of angular momentum has to be mediated by the electromagnetic fields. But what happens if the larger electromagnet is destroyed before, due to the finite speed of light, it has had a chance to "see" the brief field generated by the smaller one? I use the term destroyed to highlight the fact that not only is it not generating a field any more but that it has been rendered incapable of responding to one. How that is done is an engineering problem but not, I believe, an impossible one - for example, how about a temperature induced change in the wires of the coil from a superconductor to an insulator?

Just to make life interesting, in this thought experiment just after the smaller electromagnet has pulsed it also has its electromagnetic properties "destroyed" so it too is incapable of having its rotation changed significantly by any magnetic field that may be about.

Question: What is the carrier for the angular momentum needed to equalise that present in the (former) smaller electromagnet which has been left rotating at the end of all this "destruction"? I can only think of four possibilities:

1. Electromagnetic radiation (photons)
   
2. A remnant (orphaned) magnetic field
3. There is some other mechanism I haven't thought of which can transfer angular momentum to or from the matter that made the (former) electromagnets
4. There is a theoretical reason why the thought experiment cannot be carried out as described

I'm thinking the first option would need a lot of photons! Are there such things as orphaned magnetic fields which can carry angular momentum? Whichever option is correct the answer should be educational. :)

As mentioned, I don't think equations are needed to answer this qualitatively in the same spirit that we don't need equations to qualitatively say that magnets can attract or repel each other. Thanks for your help.

Bob.

P.S. This thread is not an invitation for replies that qualify as "Crackpottery", as the forum terms and conditions so quaintly put it. If you are tempted down that route then please post elsewhere as I don't want this thread to attract the righteous ire of the mods. There's some physics at work here that I am seeking help in understanding. Thanks again.

 

[DEvens]

Um... You are making a lot of assumptions in your "thought experiment." It isn't clear there is a problem at all. Electromagnetism can transmit force, so it can of course transmit angular momentum. It can also carry angular momentum. If you examine the equations you will find it all very nicely conservative. Angular momentum, linear momentum, energy. A very conservative force. Your "thought experiment" is no different in principle from using string or springs or jets of water.

 

[Bob not Alice]

Hi DEvans,

Thanks for chipping in. Your point about springs is well taken as they are, after all, just another expression of electrostatic forces, as is magnetism when one factors in special relativity. I agree that there isn't a problem except, perhaps, my struggle to visualise the physical processes at work.

I guess the nub of my question is how the angular momentum carried by electromagnetism is returned to the wider universe as the magnetic field(s) collapse when the electromagnets that created those fields can no longer respond to them. In the thought experiment the two electromagnets had to be separated significantly to allow the "destruction" I referred to to occur in a reasonable time span.

My best guess is that as the combined magnetic field collapses then, given it has no coils to return energy or that angular momentum to, both energy and angular momentum must be given back via photon emission as I'm unaware of any other mechanism. If that is what is occurring then is the angular momentum carried via circular polarisation?

Sorry to reward your post with more questions.

Bob.

 

[DEvens]

You still make assumptions.

Photons do indeed carry angular momentum. They do indeed carry it as spin. But they need not do it that way in order to accommodate this particular situation. More on that in one moment.

First, you need an estimate of how much angular momentum has to be "returned to the wider universe." To do that, you need to work out how much angular momentum is present that is not in either electromagnet. So when the magnetic field applies forces to the two metal objects, how much angular momentum is "in transit" so to speak at any given moment? And if the electric circuits are then broken somehow so that this magnetic field's angular momentum can't go where it would have gone, how much is actually going to have to go someplace else? Before you worry about how it is going to be carried away, think about how much has to be carried away.

To do that you will need to make some estimates about delays in transmission of fields, how much energy is in those fields, and how long it takes to get from one metal thing to the other.

What you will find is, not very much. You can get an idea of how "not very" this is by considering how much back-force there is on an electromagnet when it is activated in isolation. And that turns out to be "not very much" indeed. Remember the linear momentum carried by electromagnetic radiation is E/c. So if you had 3E8 watts of electromagnetic radiation you would get 1 Newton of force.

And that brings us back to the more likely way that this (exceedingly tiny) amount of momentum gets carried off. If the electromagnet radiates photons from one end then it produces a torque. A very tiny torque. An E/c kind of torque.