Two spheres of radii $r_{1} and r_{2}$ have densities $ρ_{1} and ρ_{2}$ and sp. heat capacities $C_{1} and C_{2}$ respectively. If they are heated to same temperature, the ratio of their rates of fall of temperature in the same surroundings will be

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$\frac{r_{2} ρ_{2} c_{2}}{r_{1} ρ_{1} c_{1}}$

$\frac{r_{1} ρ_{1} c_{1}}{r_{2} ρ_{2} c_{2}}$

$\frac{r_{1} ρ_{2} c_{2}}{r_{2} ρ_{1} c_{1}}$

$\frac{r_{2} ρ_{1} c_{1}}{r_{1} ρ_{2} c_{2}}$

answer is D.

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Detailed Solution

As per Stefan's law, $m c \frac{d θ}{d t} = - σ ε (4 {πr}^{2}) (θ - θ_{0}^{4})$

$(\frac{4 {πr}^{3}}{3} ρ) c \frac{d θ}{d t} = - σ ε (4 {πr}^{2}) (θ - θ_{0}^{4})$

$\frac{d θ}{d t} \propto \frac{1}{r c ρ}$

So required ratio $= \frac{r_{2} c_{2} ρ_{2}}{r_{1} c_{1} ρ_{1}}$

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