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 $p=\frac{1}{3}\bigg(\frac{U}{V}\bigg)$ 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
$TV ^{\gamma-1}= K$
$TV ^{\frac{\gamma}{4}}= C$
$\gamma-1=\frac{\gamma}{4}$
$\frac{3 \gamma}{4}=1$
$\gamma=\frac{4}{3}$
$TV ^{\frac{\gamma}{4}}= C$
$T V ^{\frac{1}{3}}= C$
$T\left(\frac{4}{3} \pi R^{3}\right)^{\frac{1}{3}}=C$
$T \propto \frac{1}{ R }$