Suppose the sun expands so that its radius becomes $100$ times its present radius and its surface temperature becomes half of its present value. Then, the total energy emitted by it in the same time will increase by a factor of

$[$ neglect surrounding temperature $]$

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Solution

Power of radiation from the sun,

$P = \frac{E}{t} = σ A e_{r} T^{4}$

$[$ neglecting surrounding temperature $]$

For sun, as it is an example of perfect blackbody, $e_{r} = 1$

So, $P = \frac{E}{t} = σ A T^{4}$

$\Rightarrow E = σ A T^{4} t$

In the same time, $E \propto A T^{4}$

So, $\frac{E_{2}}{E_{1}} = \frac{A_{2} T_{2}^{4}}{A_{1} T_{1}^{4}}$

$\Rightarrow \frac{E_{2}}{E_{1}} = \frac{r_{2}^{2} T_{2}^{4}}{r_{1}^{2} T_{1}^{4}}$ $[∵ A = 4 π r^{2}]$

$\Rightarrow \frac{E_{2}}{E_{1}} = \frac{(100 r)^{2} (T_{o} / 2)^{4}}{r^{2} T_{o}^{4}} = 625$

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