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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 $u = \frac{U}{V} \propto T^{4}$ and pressure $p = \frac{1}{3} (\frac{U}{V})$ . If the shell now undergoes an adiabatic expansion the relation between T and r is :

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$T \propto e^{- 3 r}$

$T \propto \frac{1}{r^{3}}$

$T \propto e^{- r}$

$T \propto \frac{1}{r}$

answer is A.

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Detailed Solution

As, $P = \frac{1}{3} (\frac{U}{V})$

$B u t \frac{U}{V} = K T^{4}$

$S o, P = \frac{1}{3} K T^{4}$

$\frac{n R T}{V} = \frac{1}{3} K T^{4} [A s P V = n R T]$

$\frac{4}{3} π r^{3} T^{3} = constatnt$

$Therefore, T \propto \frac{1}{r}$

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