# Thermal equilibrium through radiation and Stefan Boltzmann

1. Jun 25, 2012

### ktoff

Hey Guys,

so I had a longish discussion with colleagues and on reddit about thermal equilibrium and the sun and how you cannot heat up anything above the surface temperature of the sun using clever mirrors and stuff.

However, somebody came up with Napkin calculations of the Stefan Boltzmann law and I cannot put my finger on why it is not applicable.

To illustrate the problem I devised a thought experiment.

You place two objects on the focal points of a vacuum filled rotational ellipsoid. The inside of the ellipsoid is a perfect mirror and the objects have the same emissivity but different sizes/surface areas. They start out at the same temperature. And finally the ellipsoid is large with respect to the objects.

Every physics instinct screams at me that the two are at thermal equilibrium and will stay that way. However, given the configuration all the power radiated by one body is absorbed by the other and vice versa. Given the Stefan Boltzmann law, one of the bodies radiates more energy than the other (P=εσAT^4) as the surface area A is different for both bodies.

If this were true, the objects would move away from thermal equilibrium with the large body heating up the small one.

This doesn't make any sense.

So the obvious guess is, you cannot just apply the Stefan Boltzmann law here. The question is: Why?

KToff

Last edited: Jun 25, 2012
2. Jun 26, 2012

### ktoff

For anyone interested, eventually I figured out where the error is (also thanks to this site: http://tierneylab.blogs.nytimes.com/2010/03/05/the-second-law-strikes-back/).

The assumption that for a sufficiently large ellipsoid all the energy from one energy hits the other is wrong. No matter how you dimension your ellipsoid, you will never focus the entire energy of a source with a finite size onto a smaller object (as pointed out in the link, not even on an equally sized object).

Thus your energy balance is not simply one Boltzmann-term of absorbed energy minus one Boltzmann term of radiated energy but (unsurprisingly) a radiation equilibrium not violating the 2nd law of thermodynamics.