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If the temperature of the sun were to increase from $T$ to $2 T$ and its radius from $R$ to $2 R$ , then the ratio of the radiant energy received on earth to what it was previously will be_____

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answer is 64.

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Detailed Solution

From stefan’s law, energy radiated by the sun per second

$E = σ A T^{4} ⇒ A \propto R^{2}$ ;

$E \propto R^{2} T^{4}$
$\frac{E_{2}}{E_{1}} = \frac{R_{2}^{2} T_{2}^{4}}{R_{1}^{2} T_{1}^{4}}$

$\frac{E_{2}}{E_{1}} = \frac{{(2 R)}^{2} {(2 T)}^{4}}{R^{2} T^{4}}$

$= \frac{{(4 R)}^{2} (16 T^{4})}{R^{2} T^{4}}$

$E_{2} = 64 E_{1}$

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