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A spherical cavity of radius R contains black-body radiation at temperature T. Treat the radiation as a photon gas with energy density u = U/V proportional to T⁴ and pressure p = (1/3)(U/V). If the cavity expands adiabatically, how are T and R related?

1. T is inversely proportional to R
2. T is inversely proportional to R cubed
3. T is inversely proportional to R squared
4. T decreases as e^(−3R)

Correct answer: T is inversely proportional to R

Solution

For a photon gas p = u/3 with u proportional to T^4, adiabatic expansion gives T*V^(1/3) = constant. Since V is proportional to R^3, T is inversely proportional to R.

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