Re: A rotating disk PARADOX??
From: sal (believer_at_nospam.org)
Date: 06/29/04
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Date: Tue, 29 Jun 2004 14:05:41 -0400
On Tue, 29 Jun 2004 13:54:16 -0400, Pmb wrote:
>
> "sal" <believer@nospam.org> wrote in message
> news:pan.2004.06.29.15.37.57.752826@nospam.org...
>> No, I'm sure it's not really a paradox :-) . But in this variation on
>> the rotating-disk problems that were discussed here recently, I don't
>> see how angular momentum is conserved:
>>
>> We have a solenoid consisting of a single loop of (superconducting)
>> wire, generating a magnetic field. A metallic disk is in the field,
>> perpendicular to it, and it's being driven at constant rotational
> velocity.
>>
>> We connect wires to the middle and edge of the disk, using sliding
>> contacts. There is an induced voltage between the middle and edge of
>> the disk, as, I believe, is well known (and even more or less agreed to
>> in this newsgroup). We run the wires from the disk to a resistor, and
>> let current flow and heat up the resistor.
>>
>> In case this description isn't clear, here's a crude illustration of the
>> setup:
>>
>> http://physicsinsights.org/images/rotating_disk_with_solenoid.png
>>
>> The energy to heat the resistor comes from the motor driving the disk,
>> which must exert torque on the disk. Hence, the disk is also exerting
>> an equal and opposite torque on the motor, and there is a net transfer
>> of angular momentum from the disk to the Earth.
>>
>> Angular momentum must be conserved. Therefore, there must be a torque
>> caused by the rotating disk, which is acting on the _current ring_, and
>> which is equal to that applied by the motor to the shaft.
>>
>> How is that torque being generated -- what's happening to cause a
>> twisting force on the wire?
>>
>> I sure can't picture it!
>>
>> Note that any B and/or E fields generated by the rotating disk must be
>> stationary in the FoR of the ring -- i.e., the values of E and B are
>> constant in time at each point in space, since the current distribution
>> in the disk is constant in time in the frame of the ring. So, among
>> other things, the angular momentum isn't just being radiated away!
>>
>> Any E field from the disk must be purely radial (I think?) and I'm left
>> with electrons flowing in a circle which are somehow feeling a force
>> parallel to their line of motion as a result of a magnetic field, which
>> doesn't make any sense.
>>
>> An explanation of where the balancing torque comes from will be most
>> welcome.
>
> I dunno but I'll take a wild guess.
>
> Since the charges in the disk experience a magnetic force in the radial
> direction they will have a component of velocity in the radial direction.
Yes, that generates the current which is drawn off through the wires.
> But since the disk is still turning there will also be a magnetic force
> perpendicular to the radial direction.
Yes, I think the electrons follow a spiral path as they move across
the disk. I haven't made any attempt to solve it quantitatively, though,
and I have no idea what happens if the disk is a superconductor :-)
> That force constitutes a force on
> the disk.
Yes, but...
There is a current flowing in the wires and resistor. This is known.
Therefore there is energy coming out.
Therefore we _know_ that the motor must apply a torque to the shaft,
because there's no other energy going in. So, we don't really need to
worry about computing the forces on the _disk_ -- we know the net result
already, by energy arguments, and we know the general cause (qVxB), so we
can skip the messy integrals needed to figure out how the electrons on the
disk actually move until and unless we want an exact solution.
But that still leaves us with gross a torque imbalance -- torque goes in
from the motor through the shaft, so torque must come out someplace else
or we don't conserve L.
Anyhow, I think Simplicio has posted the right answer -- I can't respond
to it yet, though, for uninteresting reasons involving my news server
accounts.
-- To email me directly, take out nospam and put back physicsinsights.
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