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I'm having trouble figuring out why this following example of shenanigans wouldn't work. If you take a parabolic mirror (say from polished aluminium) with a magnification of M and construct it such that the object plane is completely encapsulated by the parabola itself. (See figure bellow)

Parabolic Mirror

Then all the light radiating from the object will be focused onto the image plane. If you place an object at the image and assume equilibrium you would get that the radiation from the objects area would equal the radiation from the image area (which has two sides, hence the factor 2): $$\pi r_o ^2 \sigma T_o^4 = 2 \pi r_i ^2 \sigma T_i^4 $$ Solving for $T_i$ gives you: $$ T_i = \sqrt{\frac{r_o}{\sqrt{2} r_i}} T_o $$

If we plug in some numbers, let's say the object is 4$\sqrt{2}$ times the size of the image, we get an interesting relationship:

$$ T_i = 2 T_o $$

Which means we can get a temperature gradient between the image and the object, and thus extract work from it. Which makes no sense, thus my question is, what's going on here?

Some assumptions I've made which might be wrong:

When an object is in focus by a lens/mirror all of the light emitted from the object which hits the lens/mirror gets focused onto the image.

The mirror is assumed to be ideal, since a less than ideal mirror can be compensated for by simply adjusting the magnification and still get the same temperature gradient.

The object and "image" are near black bodies.

So which of my assumptions are incorrect or am I doing something else wrong here?

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I realized I was doing a trivial mistake. The lens formula is only a simplification using the small angle approximation. This case is very far from small angles, thus the assumption that all light emitted from the object will be collected again at the image is wrong.

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