Question

A solid spherical black body of radius $r$ and uniform mass distribution is in the free space. It emits power $P$ and its rate of cooling is $R$, then:

• A. $RP\propto r^2$
• B. $RP\propto r$
• C. $RP\propto \frac{1}{r^2}$
• D. $RP\propto \frac{1}{r}$

Answer 1

The correct option is B $RP\propto r$

We know that rate of readiation heat transfer per unit area is proprtional to $T^4$.

$P\propto A{T}^4$ ($P$ is radiation power )

Also surface area, $A\propto r^2$
$\Rightarrow P\propto r^2.....\left(i\right)$

The power emitted by body through radiation is responsible for its cooling.
$P=ms\left(\frac{dT}{dt}\right).....\left(ii\right)$
$\frac{dT}{dt}$ is rate of cooling of body

From equation $\left(i\right)\text{and}\left(ii\right)$:
$ms\frac{dT}{dt}\propto r^2$

$\Rightarrow \frac{dT}{dt}\propto \frac{r^2}{m}$
$[\because$ $s$ is constant for body & $m=\rho V$ i.e $m\propto r^3,V=\frac{4}{3}\pi r^3]$

or, $\frac{dT}{dt}\propto r^{-1}$

Given $\frac{dT}{dt}=R$
$\Rightarrow R\propto \frac{1}{r}.....\left(iii\right)$
Now from (i) & (iii):
$PR\propto (\frac{1}{r}\times r^2)$
$\therefore PR\propto r$

$\begin{array}{c}\text{Why this question}?\\ \text{Tip: The area responsible for heat transfer through}\\ \text{radiation is surface area of spherical body i.e}A=4\pi r^2.\\ \text{The rate of heat transfer through radiation can be related as}\\ \frac{dQ}{dt}=eA\sigma T^4\text{or}\frac{dQ}{dt}\propto A{T}^4\end{array}$