The Circle-ellipse paradox: a problem in thermodynamics

Circle-Ellipse ParadoxThis puzzle is represented here in two dimensions, but would be the same in three (in which case think ellipsoid of revolution and sphere - or, alternatively, imagine a long pipe with the given cross-section, extending out of and into the screen).

On the left of the drawing is a an ellipse, which has been cut off in the example somewhere between its centre and the right-hand focus B. A matching cut-off circle is attached, whose centre B corresponds with the right-hand focus of the ellipse. Objects are placed at the foci A & B. The interior surface of the circle-ellipse casing is reflecting - and we imagine the reflection to be perfect.

Now the geometry of the ellipse guarantees that any radiation leaving focus A and reflecting directly from the ellipse will arrive at B (red line 3). Similarly, any radiation leaving B and reflecting directly from the ellipse will arrive at A (red line 2). For the circle, any radiation leaving B and impinging directly on the interior of the circle will reflect back to B (red line 1). It is not quite so obvious what will happen to radiation heading from A to the circle's interior surface (red line 4), but by elimination it is possible to see that it must eventually return back to A (because all that returns to B is already accounted for).

It's clear, from the relative sizes of the angles in the picture, that when A and B are at the same temperature then more radiation must travel from A to B than from B to A.
To elaborate, let us take some example figures and say that:
 - from A, about 80% reflects to B, and 20% must eventually return to A
 - from B, about 65% reflects back to B, and 35% to A
(don't waste time measuring the angles in the diagram - the exact percentages are not important)

Assuming that A and B start off at the same temperature, then B must heat up while A cools down. But this breaks the second law of thermodynamics. In other words, it counts as a perpetual motion machine - and it is a well-known fact that no such devices work.

Of course, one doesn't see "motion" as such in this device. But imagine a long pipe with a smooth & silvered interior surface of the circle-ellipse cross-section, along the interior of which two thin water tubes are placed at the focus positions. Cold water would be fed into tube A, and hot water could be taken from tube B. A zero-energy heat-pump!

PLEASE DO NOT waste your time building such an apparatus - it won't work! The question is - - - - why not?

* * * * * * * * *

There is an analogous paradox in acoustics: Eisner's Reciprocity Paradox. Eisner imagined an ellipse with one end cut off - i.e. like the above but with no circle attached. There is a principle in acoustics that the sound heard at B from a source at A should be the same as the reverse* (I hope I have paraphrased that adequately); but how so? most of a sound emitted at B would dissipate, while it's quite reasonable that most of a sound emitted at A would get to B. Perhaps this Stanford website throws some light on it.

         * so be careful what you say in a concert hall!

The flaw in the circle-ellipse paradox can be answered very simply, without a lot of mathematical detail.

John Kilpatrick                Home Page       Email        Latin curiosities    Wordsquares
Feb. 2011