Question

Spheres $P$ and $Q$ are uniformly constructed from the same material which is a good conductor of heat and the radius of $Q$ is thrice the radius of $P$. The rate of fall of temperature of $P$ is $x$ times that of $Q$. When both are at the same surface temperature, the value of $x$ is :

• A. $\frac{1}{4}$
• B. $\frac{1}{3}$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

From Newton's law of cooling,

$\frac{dT}{dt}=-\frac{4\sigma eA{T}_0^3}{ms}(T-T_0)=-\frac{4\sigma e\left(4\pi r^2\right)T_0^3}{\left(\rho 4/3\pi r^3\right)s}(T-T_0)$
Thus, $\frac{dT}{dt}\propto \frac{1}{r}$
$\frac{{\left(\frac{dT}{dt}\right)}_P}{{\left(\frac{dT}{dt}\right)}_Q}=\frac{r_Q}{r_P}=\frac{3r_P}{r_P}=3$
$\therefore {\left(\frac{dT}{dt}\right)}_P=3\times {\left(\frac{dT}{dt}\right)}_Q$