[This question is Certified Higgs Free!]
Richard Feynman in Lectures on Physics Vol. II Sec. 17-4, "A paradox," describes a problem in electromagnetic induction that did not originate with him, but which has nonetheless become known as "Feynman's Disc Paradox." It works like this: A disc (Feynman's spelling) that is free to rotate around its axis has set of bead-like static charges near its perimeter. The disc in it also has a strong magnetic field whose north-south axis is parallel to the rotation axis of the disc. The disc with its embedded static charges and magnetic field is initially at rest.
However, the magnetic field is was generated by a small superconducting current. The disc is permitted to warm up until the magnetic field collapses.
The paradox is this: Conservation of angular momentum says that after the field collapses, the disc must of course remain motionless. However, you could also argue that since the collapsing magnetic field will create a strong circular electric field that is tangential to the perimeter of the disc, the static charges will be pushed by that field and the disc will necessarily begin to rotate.
Needless to say, you can't have it both ways!
Feynman, bless his heart, seemed to have an extraordinarily optimistic view of the ability of others to decipher some of his more cryptic physics puzzles. As a result, I was one of many people who years ago discovered to my chagrin that he never bothered to answer his own question, at least not in any source I've ever seen.
In the decades since then, that lack of resolution has produced a surprisingly large number of published attempts to solve the Feynman Disc Paradox. Many of these are summarized in a paper that was written and updated a decade ago by John Belcher (MIT) and Kirk T. McDonald (Princeton) (Warning: I can see the paper, but it may have access restrictions for others.)
My problem is this: I more-or-less accidentally came up with what seems to be a pretty good resolution of the paradox, and it ain't the one described in any of the papers I've seen on it. But I can't easily back off, because the solution is a bit too straightforward once you look at it the right way. I think!
I also think that Feynman's solution was very likely to have been relatively simple, and not some kind of tremendously detailed exercise in relativistic corrections. He was after all trying to teach freshmen, and he honestly seemed to think they would all figure it out with a bit of thought!
So, help me out here folks: Does anyone know for sure what Feynman's solution to this little puppy was? Along those lines, is Laurie M Brown from Northwestern by any chance linked into this group? I can't imagine anyone who knows more about Feynman's published works!
I will of course explain why I think there's a simple solution, but only after seeing if there's something simple (but apparently hard to find) already out there.
Addendum: The Answer!
I am always delighted when a question can be answered so specifically and exactly! @JohnMcVirgo uncovered the answer, right there in Volume II of the Feynman Lectures... only 10, count 'em 10, chapters later, in the very last paragraph of FLoP II 27, in Section 27-6 ("Field Momentum"), p 27-11:
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.
Feynman hints at the above answer in earlier chapters, but never comes right out with a direct reference back to his original question.
John McVirgo, again, thanks. I'll review FLoP II 27 in detail before deciding whether to post that "other viewpoint" I mentioned. If Feynman already covers it, I'll add another addendum on why I think it's important. If the viewpoint is not clear, I'll need to do some simple graphics to explain how it may add some clarity to how the angular momentum conservation part works.
Addendum 2012-07-08: Not The Answer!
In the comments, @JohnMcVirgo has very graciously noted that I read more into his answer than he had intended, and for that reason he did not feel he should receive the answer mark. By finding that bit of text at the very end of the chapter John mentioned, I may in fact have answered my own question, at least in the literal sense of "what did Feynman say about it?" But John also points out his own surprise on how Feynman answered, which is different from points made by both him and @RonMaimon. So for now I'm leaving this question open. I will assign an answer eventually, but only after I've read up on FLoP II 27 to the point where I feel I know it inside out.
Addendum 2012-07-08: New Answer!
Well that was a short several weeks! @RonMaimon's additions to his initial answer, combined with his latest comment clarifying the difference between field momentum and "mechanical" momentum, demonstrate a deep understanding of the issues. Since @JohnMcVirgo already suggested the updated Ron Maimon text as the answer, I agree and have so designated it. I am still deeply grateful to John for pointing me to FLoP II 27, since without that clue I never would have found Feynman's answer in his own words.
I will at some point bring up my "other view" of Poynting problems as a new question. I now have two of those pending, since I am also still planning an updated Dual Cloud Chamber problem at some point.
No comments:
Post a Comment