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Two spherical black bodies of radii $r_{1}$ and $r_{2}$ and with surface temperature $T_{1}$ and $T_{2}$ respectively radiate the same power. Then the ratio of $r_{1}$ and $r_{2}$ will be

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${(\frac{T_{2}}{T_{1}})}^{2}$

${(\frac{T_{2}}{T_{1}})}^{4}$

${(\frac{T_{1}}{T_{2}})}^{2}$

${(\frac{T_{1}}{T_{2}})}^{4}$

answer is A.

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Detailed Solution

$P = A σ T^{4} = 4 π r^{2} σ T^{4} \Rightarrow P \propto r^{2} T^{4}$

$r^{2} \propto \frac{1}{T^{4}}$ [As P=constant] $∴ \frac{r_{1}}{r_{2}} = {(\frac{T_{2}}{T_{1}})}^{2}$

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