First slide

Radiation

Question

Two metallic spheres $S_{1} and S_{2}$ are made of the same material and have identical surface finish. The mass of $S_{1}$ is three times that of $S_{2}$ . Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of $S_{1}$ to that of $S_{2}$ is

Moderate

A

$\frac{1}{3}$
B

${(\frac{1}{3})}^{\frac{1}{3}}$
C

$\frac{1}{\sqrt{3}}$
D

$\frac{\sqrt{3}}{1}$

Solution

Rate of cooling (R) = $\frac{∆ θ}{t} = \frac{A \in_{0} σ (T^{4} - {T_{0}}^{4})}{mc}$
$\Rightarrow R is directly proportional to \frac{A}{m} directly proportional to \frac{Area}{Volume} directly proportional to \frac{r^{2}}{r^{3}} = \frac{1}{r}$
$\Rightarrow Rate (R) directly proportional to \frac{1}{r} directly proporitonal to \frac{1}{m^{\frac{1}{3}}}$
[m = $ρ \times \frac{4}{3} {πr}^{3} \Rightarrow r directly proportional to m^{\frac{1}{3}}]$
$\Rightarrow \frac{R_{1}}{R_{2}} = {(\frac{m_{2}}{m_{1}})}^{\frac{1}{3}} = {(\frac{1}{3})}^{\frac{1}{3}}$

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