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 of 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 ;

Answer 1

Given,

$R_Q=3R_P$

From Newton's law of cooling,

$\frac{dT}{dt}=-\frac{4\sigma eA{T}_o^3}{ms}(T-T_o)=-\frac{4\sigma e\left(4\pi r^2\right)T_o^3}{\left(\rho \frac{4}{3}\pi r^3\right)s}(T-T_o)$

Thus, $\frac{dT}{dt}\propto \frac{1}{r}$

$\Rightarrow \frac{{\left(\frac{dT}{dt}\right)}_P}{{\left(\frac{dT}{dt}\right)}_Q}=x=\frac{R_Q}{R_P}=3$.

Thus the value of x is 3.