Feynman's Paradox: Solving the Mystery of the Rotating Disk | Solution Revealed

[availablegod]

hi folks,
this is my first post..and here i would like to present a solution to a paradox posed by feynman in the 2nd volume of his lectures (17-4)...for those of you who wish to see the problem here's the link http://varatek.com/scott/feynman_problems.html
it's the last problem in that page.
Here's my solution :
Considering that there is no other object except the ‘Feynman disc’ in a particular universe, the initial energy stored is the sum of the magnetic field energy of the coil, the potential energy of the system of the charged spheres and the potential energy contained in the mechanical structure itself. When the current is zero then there is zero magnetic field energy in the coil. Now if the disk doesn’t rotate then we can clearly see that there is a violation of the principle of energy conservation. Hence the only other option left for the disk is to rotate and account for that missing energy by the virtue of its rotational kinetic energy.

Since we have figured out that the disk will be actually rotating using the most fundamental conservation law in this universe (Namely the principle of conservation of energy), we assume that there must be some angular momentum already existing in the stationary system. We can logically assume that the angular momentum could possibly be stored in either of the static electromagnetic or electrostatic field, because this is where the energy of the system is initially stored (anything which has an angular momentum is a source of energy). Since the angular momentum that is initially stored in either of these two, is ultimately transferred to the disk, hence we can safely assume that the angular momentum is stored in the static electromagnetic field (the one being ‘changed’). And thus the solution to this whole paradox can be stated as “a static electromagnetic field has an angular momentum attached to it”. Which in fact is the case as have been found by Stedman (1981) and Calkin (1966).

love it or loathe it...just let me know

 

[kcdodd]

Your conclusion is correct, but why assume that the energy must go into the disk for it to be conserved. It could all just radiate away. Depending on how massive the disk is it will have different final kinetic energy.

 

[availablegod]

actually I have considered a universe which does not contain any'thing' other than the apparatus being considered. I have considered the following facts
1) the answer to this paradox does not depend on the dimensions of the universe being chosen
2) the answer should also not depend on the presence of other objects in that universe.
Now suppose that I choose a universe whose boundaries coincide with the boundaries of the feynman disc apparatus...Now my question is that 'can the energy radiate out without doing any work'? because surely the energy cannot radiate out of the universe because there will be a violation of the energy conservation principle

 

[kcdodd]

If there was no disk at all, the energy would still go somewhere in your "limited universe" model.

Also, you must consider the possibility that there was no momentum in the field to begin with. Conservation simply states everything adds up the same. So, if you start with zero, you can end with zero by adding to the disk in one direction, and adding to the field in the other direction.

 

[availablegod]

if there were no disk at all..then where would the energy come from? Would you please explain the second part of your last post in a little bit more detail? I really want to understand your statement

 

[codelieb]

Feynman addresses this (so-called) paradox in the final paragraph of FLP Vol. II Chapter 27, Field Energy and Field Momentum:

"Finally, another example is the situation with the magnet and the charge,
shown in Fig. 27-6. We were unhappy to find that energy was flowing around
in circles, but now, since we know that energy flow and momentum are proportional,
we know also that there is momentum circulating in the space. But
a circulating momentum means that there is angular momentum. So there is
angular momentum in the field. Do you remember the paradox we described in
Section 17-4 about a solenoid and some charges mounted on a disc? It seemed
that when the current turned off, the whole disc 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."

This, however, is merely a qualitative description; sufficient quantitative (mathematical) information about the energy and momentum of the electromagnetic field is given in this chapter that a Caltech sophomore in his second year of the Feynman Lectures course would be expected to be able to find the final angular frequency of the disc as a function of the various parameters it depends on.

Mike Gottlieb
Editor, The Feynman Lectures on Physics, Definitive Edition
---
Physics Department
California Institute of Technology

 

[availablegod]

I am honoured that you took time out to reply to this thread..thank you professor Gottlieb.
btw do you think that my line of reasoning is flawed?

 

[kcdodd]

if there were no disk at all..then where would the energy come from? Would you please explain the second part of your last post in a little bit more detail? I really want to understand your statement

You stated that the solenoid has an initial current and magnetic field, which means there is energy stored in the field.

My second claim is Newtons third law. Imagine a stationary object exploding. All the pieces have momentum, but they must all add up to zero because the initial object had zero momentum.

 

[availablegod]

You stated that the solenoid has an initial current and magnetic field, which means there is energy stored in the field.

My second claim is Newtons third law. Imagine a stationary object exploding. All the pieces have momentum, but they must all add up to zero because the initial object had zero momentum.

energy is stored in the field but it is only because the solenoid is there..the disc's dimensions are not important here...the disc might have a diameter equal to that of the solenoid which means that it is enough for the system boundary to just cover the solenoid only. I have approached the problem in this manner : 1) Find out what would happen using conservation of energy 2) use that data to predict the property of the field
then why would I have to consider the possibility that there might be zero angular momentum in the field initially. Correct me if I am wrong :)

 

[kcdodd]

Imagine an infinite solenoid with no field outside. Surround it with a concentric infinite cylinder of charge with no field inside. The electric and magnetic fields never cross in the static case, which means there is no angular momentum in the field. But cause the current to collapse in the solenoid and you still have an electric field outside which will cause the external cylinder to rotate.

 

[availablegod]

thanks a lot..i blundered in the 'momentum' part..thanks for pointing out :-)

 

[vin300]

The angular momentum isn't zero after all.There's the angular momentum of the electrons.If you're in the frame of the electons, there's angular momentum of the disk.If the disk is made bigger with the spheres farther, the electric field at the charges become smaller with increse in M.I., lesser force such that torque remains constant

 

[kcdodd]

There may or may not be any mechanical angular momentum in the solenoid itself, like from electron movement. There is no direct relation between current and momentum, other than a qualitative statement that there is one. Either way, the solenoid is neutral and rigid. The only possible em force on the solenoid is through its current. JxB would be a radially symmetric force and thus not contribute to angular momentum.