Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume u = U/V ∝ T⁴ and pressure p = (1/3)(U/V). If the shell now undergoes an adiabatic expansion the relation between T a

Question

Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume u = U/V ∝ T⁴ and pressure p = (1/3)(U/V). If the shell now undergoes an adiabatic expansion the relation between T and R is-

Accepted Answer

T ∝ 1/R. During an adiabatic expansion of the photon gas, the internal energy decreases as the volume increases. Since the internal energy is proportional to the temperature raised to the fourth power (u ∝ T⁴), and the volume is proportional to the cube of the radius (V ∝ R³), the relationship between temperature and radius simplifies to T ∝ 1/R.

Solution

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