EM fields. Angular Momentum seems not to be Conserved

[FedEx]

EM fields. Angular Momentum seems not to be Conserved!

Hi everyone..

This is a paradox which as been stated in Feynman Lectures 2 (Chapter 17-4).

What Feynman is telling is, there is a plastic plate which is free to rotate about an axis passing through the center of the disc, there are several metal spheres having a charge Q which are sticked to the plate. Now at the centre of the plate there is a current carrying solenoid of some length put in a way that the axis of the solenoid coincides with the axis of the disc. The solenoid is made of a supercondcuting material. And the temp is right now let's say 0.1 K.

Now as the temp increases the superconductivity is lost and hence the current is lost. So the flux of B decreases(for an Amperian loop concentric to the solenoid) and hence that creates an E which will try to reduce the decrease in the flux. Now this E inturn applies a force to the charged metal spheres.

Hence there is a net force on the plate. Hence the disc rotates. The disc has some angular momnetum. But intially the angular momentum is zero. So what's happening? Is the law of conservation of momentum wrong?

My thoughts-
Law of conservation is a beautiful truth of nature. It cannot be wrong.

There are two outcomes.
1)The plate doesnot rotate. Then momentum is conserved. But then why does the disc doensot rotate?

2)The plate does rotate. In this case what i have come to think is that. If i change the charge of the spheres to -Q from +Q. The final angular momentum will have a changed direction. And now as my initial momentum should be equal to final. My inital momentum s thus depndent on the charge of the metal spheres. And this is absurd. How can the Initial momentum depend on the charge of the spheres unless there is something like the momentum of the fields which interacts with the spheres and stuff like that...

So help me with it. Dont give the answer. Give me a hint or an argument which might help me get out of this wonderful paradox.

 

[Dale]

You probably know that an EM field has energy and momentum, but what you may not realize is that it also contains angular momentum. So, whatever angular momentum is gained by the system is lost by the fields.

Here is a good starting point, but it is pretty much a spoiler for your question, so you may want to start elsewhere if you don't want an answer:
http://physics.unipune.ernet.in/~phyed/23.4/23.4_Curiosities-5.pdf

 

[FedEx]

You probably know that an EM field has energy and momentum, but what you may not realize is that it also contains angular momentum. So, whatever angular momentum is gained by the system is lost by the fields.

Here is a good starting point, but it is pretty much a spoiler for your question, so you may want to start elsewhere if you don't want an answer:
http://physics.unipune.ernet.in/~phyed/23.4/23.4_Curiosities-5.pdf

I guess the article is solely dedicated to the problem which feynman has given.. Will look through it..

I still haven't considered topics like momentum of fields, will read them and then go for the problem..

 

[kcdodd]

As DaleSpam says, the field itself carries momentum. If the plate gains angular momentum from the field, the field will gain angular momentum from the plate in the opposite direction. This is basically an extension of Newtons 3rd law.

 

[GRDixon]

Hi everyone..

Dont give the answer. Give me a hint or an argument which might help me get out of this wonderful paradox.

The original current in the coil has nonzero angular momentum. What happens when the current in the coil is reduced to zero by Ohmic forces? Hint: you don't need the charged metal spheres. Rotation of the plate will also occur if the coil is glued to the plate.

I suspect the principle Feynman is referring to is self-inductance. Part and parcel of that is the fact that the coil's angular momentum, at a given current, will depend on the coil's geometry (e.g. its radius) and the number of wire loops per unit length in the coil.

 

[kcdodd]

There is no direct conversion from current to momentum. You may as well take the momentum of the current itself to be zero.

 

[FedEx]

As DaleSpam says, the field itself carries momentum. If the plate gains angular momentum from the field, the field will gain angular momentum from the plate in the opposite direction. This is basically an extension of Newtons 3rd law.

Newton's third law?... Well i am not sure about that.. When it comes to the questions in which there are no force fields acting then you i can say that for a given two bodies if one gets r cross f then the other gets r cross minus f... But i don't know whether you can apply that to the fields..

 

[Dale]

Newton's 3rd law is the conservation of momentum in Newtonian mechanics. So any conservation of momentum principle is essentially a generalization of the 3rd law.

 

[FedEx]

The original current in the coil has nonzero angular momentum. What happens when the current in the coil is reduced to zero by Ohmic forces? Hint: you don't need the charged metal spheres. Rotation of the plate will also occur if the coil is glued to the plate.

I suspect the principle Feynman is referring to is self-inductance. Part and parcel of that is the fact that the coil's angular momentum, at a given current, will depend on the coil's geometry (e.g. its radius) and the number of wire loops per unit length in the coil.

I don't think so... I thought the same but then thought it was not fitting.. Cause the final momentum depends on the CHARGE of the spheres.. While you are considering the momentum of the electrons which in the first case is too small.. And also final momentum is greater if the rate at which the current dies down is faster.. While the model which you are saying doesnot seem to encapsulate any of these arguments..

 

[FedEx]

Newton's 3rd law is the conservation of momentum in Newtonian mechanics. So any conservation of momentum principle is essentially a generalization of the 3rd law.

Indeed... It is nothing but the angular momentum conservation.. What i meant to say that is with the fields it is not so trivial like r cross miuns f gives rate of change of field mometum.. There is some more physics (and math) to it i guess..

 

[GRDixon]

While you are considering the momentum of the electrons which in the first case is too small

"…the first thought that arises in the mind is that the electricity circulates with something like momentum or inertia in the wire." J.C.Maxwell, "A Treatise on Electricity and Magnetism". I believe that Maxwell's initial take was correct. The angular momentum is not simply the sum of the individual electrons' momentum. The electrons interact with one another when they constitute a current in a coil, and when the current is varied in time. The net torque that the coil exerts, when the current is varied, is the sum of that attributable to the electrons' inertial (self) forces, plus these myriad interactive forces.

 

[FedEx]

"…the first thought that arises in the mind is that the electricity circulates with something like momentum or inertia in the wire." J.C.Maxwell, "A Treatise on Electricity and Magnetism". I believe that Maxwell's initial take was correct. The angular momentum is not simply the sum of the individual electrons' momentum. The electrons interact with one another when they constitute a current in a coil, and when the current is varied in time. The net torque that the coil exerts, when the current is varied, is the sum of that attributable to the electrons' inertial (self) forces, plus these myriad interactive forces.

But then what about the case when the charge on the spheres is -Q. Or let's say that the spheres are neutral... The final momentum is different in both the cases.

 

[GRDixon]

But then what about the case when the charge on the spheres is -Q. Or let's say that the spheres are neutral... The final momentum is different in both the cases.

Good questions. I need to think about this harder.

 

[FedEx]

Good questions. I need to think about this harder.

Thanks. I guess there is something like the momentum of fields... I guess its impossible to solve this without reading about the momentum of fields...