Question

A hot metallic sphere of radius r radiates heat. Its rate of cooling is
A. independent of r
B. proportional to r
C. proportional to ${{r}^{2}}$
D. proportional to $\dfrac{1}{r}$

Answer 1

Hint: In this question, we are given a hot metallic sphere which is of radius r and we have to find its rate of cooling. For this we use the formula of rate of cooling and then by comparing the rate of cooling with radius, we find that the rate of cooling is inversely proportional to $\dfrac{1}{r}$.

Formula Used:
We use the formula of rate of cooling which is
${{R}_{c}}=\dfrac{d\theta }{dt}=\dfrac{A\varepsilon \sigma {{T}^{4}}}{mc}$
Where $\varepsilon $ is the emissivity of the body, $\sigma $ is the Stefan- Boltzmann constant, A is the surface of the body and T is the absolute temperature of the body.

Complete step by step solution:
We have given a hot metallic sphere of radius r. And we have to find the rate of cooling. We know the formula of rate of cooling is,
${{R}_{c}}=\dfrac{d\theta }{dt}=\dfrac{A\varepsilon \sigma ({{T}_{1}}^{4}-{{T}_{0}}^{4})}{mc}$
So from the above equation, we conclude that
$\dfrac{d\theta }{dt}\propto \dfrac{A}{V}\propto \dfrac{{{r}^{2}}}{{{r}^{3}}}$
Hence $\dfrac{d\theta }{dt}\propto \dfrac{{{r}^{2}}}{{{r}^{3}}}$
Simplifying further, we get
This means $\dfrac{d\theta }{dt}\propto \dfrac{1}{r}$
Thus from the above expression, we conclude that the rate of cooling is proportional to $\dfrac{1}{r}$.

Thus, option D is the correct answer.

Note: Radiation is a form of transfer of heat through electromagnetic radiation. This radiation carries heat energy in the form of photons from one place to another. A metallic body does the same when it gets heated. It does not require a medium of propagation.