Question

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at $0\, K$. At time $t =0$, the temperature of the object is $200 \,K$. The temperature of the object becomes $100 \,K$ at $t = t _{1}$ and $50\, K$ at $t = t _{2}$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio $\left( t _{2} / t _{1}\right)$ is______.

Answer 1

$- C \frac{ dT }{ dt }=\left( T ^{4}- T _{ s }^{4}\right)$
$\int\limits_{200}^{100} \frac{ dT }{ T ^{4}- T _{ s }^{4}}=\int\limits_{0}^{ t }-\frac{1}{ c } dt$
$-\frac{1}{3}\left[\frac{1}{ T ^{3}}\right]_{200}^{100}=-\frac{1}{ c }\left( t _{1}\right)$
$\Rightarrow \left[\frac{1}{(100)^{3}}-\frac{1}{(200)^{3}}\right]=\frac{3}{ c } t _{1}$
Similarly, $\left[\frac{1}{(50)^{3}}-\frac{1}{(200)^{3}}\right]=\frac{3}{ c } t _{2}$
$\frac{ t _{2}}{ t _{1}}=\frac{\left[\frac{1}{(50)^{3}}-\frac{1}{(200)^{3}}\right]}{\left[\frac{1}{(100)^{3}}-\frac{1}{(200)^{3}}\right]}=9$