First slide

Radiation

Question

Two spherical bodies A and B of diameter $d = 0.0344 m m$ are heated to the same temperature and allowed to coal in the same ambient temperature. Ratio of emissivities of their surfaces is 1 : 2, ratio of their specific heats is 2 : 3, ratio of their radii is 2 : 1 and ratio of their densities is 1 : 3. Then ratio of their initial rate of cooling is

Difficult

A

9 : 5
B

5 : 4
C

7 : 6
D

9 : 8

Solution

$(\frac{4}{3} π R^{3} ρ) . S . (- \frac{d θ}{d t}) = e . σ . (4 π R^{2}) (θ^{4} - θ_{0}^{4})$
$\Rightarrow R_{C} = - \frac{d θ}{d t} = \frac{3 e σ}{P S R} (θ^{4} - θ_{0}^{4})$
$∴$ Initially, $R_{c} \propto \frac{e}{ρ S R}$
$∴ \frac{{(R_{c})}_{A}}{{(R_{c})}_{B}} = (\frac{e_{A}}{e_{B}}) (\frac{ρ_{B}}{ρ_{A}}) (\frac{S_{B}}{S_{A}}) (\frac{R_{B}}{R_{A}}) = \frac{1}{2} .3. \frac{3}{2} . \frac{1}{2} = \frac{9}{8}$

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