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A paradox?!?!?!

Well, this is a paradox from Feynman lectures on Physics, so if it is right to just discuss it here, then I would like to do so.

It says that we have a device in which there is a circular plastic disc in whose center, there is an axis. Around the axis, a coil of wire is attached which is further attached to a battery. Also, on the perimeter of the disc, there are some equally charged metal spheres. Everything is at rest. Suppose now that by accident, the current in the solenoid is interrupted. So long as the current continued, there was a magnetic flux more or less parallel to the axis. When the current is interrupted, the magnetic flux must go to 0. There will therefore be an E-field induced which will circulate around in the circles centered at the axis. The charged spheres on the perimeter will all experience an E-field tangential to the perimeter. There will be a net torque since field is same in each one.

Now, there can be 2 arguments as to if the disc will rotate when the current is stopped.

  1. From above argument only, when the current is stopped, there is a change in magnetic flux and there should be an induced emf, and the disc must rotate.

  2. But, using the principle of conservation of angular momentum, the angular momentum of the system previously was 0 and it should be so when current disappears. There should be no rotation.

Now, what's the answer to this paradox?

Note by Kartik Sharma
2 years, 1 month ago

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The "counter" angular momentum is actually carried away by electromagnetic radiations ( you may be familiar that they carry a momentum equals E/c) Hence you cannot apply conservation of angular momentum unless you take the radiations into account Mvs Saketh · 2 years, 1 month ago

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@Mvs Saketh Sorry but they are carrying a momentum as you say and then an angular momentum? I didn't get that. Kartik Sharma · 2 years, 1 month ago

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@Mvs Saketh Well, your answer seems quite right to me but doesn't fascinate me a lot. Actually, this was near the same explanation that I first thought. I used the verb "fascinate" because Feynman says that "We should also warn you that the solution is not easy, nor is it a trick. When you figure it out, you will have discussed an important principle of electromagnetism"

Now, is the "important principle of electromagnetism" just that "angular momentum is carried away by electromagnetic radiations"? I have one more question if "back emf" has some role to play here or not. Kartik Sharma · 2 years, 1 month ago

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@Kartik Sharma any think that has momentum has an angular momentum (as L=rXp) , so dont worry about that :) (infact, the principle of conservation of angular momentum is a direct consequence of conservation of linear momentum, (its not a new law)) Mvs Saketh · 2 years, 1 month ago

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It seems like a paradox only because we assume that for a non-rotating disc, the angular momentum is zero. We're forgetting that the electric current in the disc has an angular momentum, which is conserved when it was converted into mechanical rotation. The work is showing them to be exactly the same. Michael Mendrin · 2 years, 1 month ago

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@Michael Mendrin Now in layman's terms, that is actually just that the electrical energy gets converted to mechanical energy, right? Kartik Sharma · 2 years, 1 month ago

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@Kartik Sharma That's right. The conceptual leap that needs to be made here is that the angular momentum of an electric current in a disc is a thing. Just because you don't see anything moving doesn't mean there is no angular momentum. Michael Mendrin · 2 years, 1 month ago

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@Michael Mendrin Oh k, I get it! But now as was expected, I would ask - what's the reason behind electric current having an angular momentum? I didn't get this. I did although assumed first that there has to be something like that only. It should have a momentum but why angular momentum? Does that mean every moving body has an angular momentum? Kartik Sharma · 2 years, 1 month ago

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@Kartik Sharma Kartik, this is a deceptively difficult subject, on which quite a few professional-grade papers have been written. But here are Feynman's own words in explaining this:

Do you remember the paradox we described in Section 17-4 about a solenoid and some charges mounted on a disc? It seemed when the current turned off, the whole disk should start to turn. The puzzle was: Where did the angular momentum come from? The answer is that if you have a magnetic field and some charges, there will be some angular momentum in the field. It must have been put there when the field was built up. When the field is turned off, the angular momentum is given back. So the disc in the paradox would start rotating. This mystic circulating flow of energy, which at first seemed so ridiculous, is absolutely necessary. There is really a momentum flow. It is needed to maintain the conservation of angular momentum in the whole world.

Apparently you need both an electric and a magnetic field for it to have a momentum, which can be either linear or angular, depending on the arrangement of the EM field. Michael Mendrin · 2 years, 1 month ago

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@Michael Mendrin Hmm Okay I got it now! Thanks a lot @Michael Mendrin! And sorry if I disturbed you in any way! I would love to know more about this though! If you get anything, please share it! Thanks again BTW! Kartik Sharma · 2 years, 1 month ago

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@Kartik Sharma Check out my comment to Mvs. Kartik Sharma · 2 years, 1 month ago

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Here's one more example of a similar paradox:

Attach two small charged spheres to the ends of an insulating spring. Take this system to space and compress the spring and let go. Now, the charges will execute oscillatory motion in which each of them is accelerated. Since they are accelerating, they emit E.M waves. And since, their total potential(spring) and kinetic energies must be conserved, the oscillatory motion never stops. We now have a system which emits E.M waves(energy) which never stops. \(\Rightarrow @ \infty\) energy!

Raghav Vaidyanathan · 2 years, 1 month ago

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@Raghav Vaidyanathan @raghav- its not a paradox, you have to incorporate relativity in order to show that the system loses energy and stops , using classical theory doesnt even predict that EM waves are released.

So, if you use a classical theory to explain a non classical, phenomenon, ofcourse, we will get paradoxes Mvs Saketh · 2 years, 1 month ago

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@Ronak Agarwal @Mvs Saketh @Raghav Vaidyanathan @Michael Mendrin and @Brilliantians Kartik Sharma · 2 years, 1 month ago

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