Question

The temperature of an spherical isolated black body falls from $T_1$ to $T_2$ in time $t$. Then time $t$ is

• A. $t\propto (\frac{1}{T_2}-\frac{1}{T_1})$
• B. $t\propto (\frac{1}{T_2^2}-\frac{1}{T_1^2})$
• C. $t\propto (\frac{1}{T_2^3}-\frac{1}{T_1^3})$
• D. $t\propto (\frac{1}{T_2^4}-\frac{1}{T_1^4})$

Answer 1

The correct option is C $t\propto (\frac{1}{T_2^3}-\frac{1}{T_1^3})$

Power radiated $P=4\pi r^2\sigma T^4=-\left(ms\frac{dT}{dt}\right)-\frac{dT}{T^4}=\frac{4\pi r^2\sigma dt}{ms}=cdt-{\int }_{T_1}^{T_2}\frac{dT}{T^4}=ct\Rightarrow t=K[\frac{1}{T_2^3}-\frac{1}{T_1^3}]$