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A spherical cavity of radius R is filled with black-body radiation at temperature T. Treat the radiation as an ideal 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 varies inversely with R
2. T varies inversely with R cubed
3. T decreases exponentially with R
4. T decreases as e to the power of minus 3R

Correct answer: T varies inversely with R

Solution

In an adiabatic expansion of a black-body radiation-filled cavity, the energy density decreases as the volume increases, leading to a decrease in temperature. Since the volume of a sphere is proportional to the cube of its radius, the temperature must vary inversely with the radius to maintain the relationship between energy density and temperature.

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