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

• A. $T\propto e^{-R}$
• B. $T\propto \frac{1}{R}$
• C. $T\propto e^{-3R}$
• D. $T\propto \frac{1}{R^3}$

Answer 1

Answer: (B)

$u=\frac{U}{V}\propto T^4$
$P=\frac{1}{3}\left(\frac{U}{V}\right)$
Adiabatic expansion
$T{V}^{\gamma -1}=K$
$T{V}^{\frac{\gamma }{4}}=C$
$\gamma -1=\frac{\gamma }{4}$
$\frac{3\gamma }{4}=1$
$\gamma =\frac{4}{3}$
$T{V}^{\frac{\gamma }{4}}=C$
$T{V}^{\frac{1}{3}}=C$
$T{\left(\frac{4}{3}\pi R^3\right)}^{\frac{1}{3}}=C$
$T\propto \frac{1}{R}$