JEE MAIN - Physics (2004 - No. 55)
If the temperature of the sun were to increase from $$T$$ to $$2T$$ and its radius from $$R$$ to $$2R$$, then the ratio of the radiant energy received on earth to what it was previously will be
$$32$$
$$16$$
$$4$$
$$64$$
Explanation
$$E = \sigma A{T^4};\,\,A \propto {R^2}$$
$$\therefore$$ $$E \propto {R^2}{T^4}$$
$$\therefore$$ $${{{E_2}} \over {{E_1}}} = {{R_2^2T_2^4} \over {R_1^2T_1^4}}$$
$$ \Rightarrow {{{E_2}} \over {{E_1}}}$$
$$ = {{{{\left( {2R} \right)}^2}{{\left( {2T} \right)}^4}} \over {{R^2}{T^4}}}$$
$$ = 64$$
$$\therefore$$ $$E \propto {R^2}{T^4}$$
$$\therefore$$ $${{{E_2}} \over {{E_1}}} = {{R_2^2T_2^4} \over {R_1^2T_1^4}}$$
$$ \Rightarrow {{{E_2}} \over {{E_1}}}$$
$$ = {{{{\left( {2R} \right)}^2}{{\left( {2T} \right)}^4}} \over {{R^2}{T^4}}}$$
$$ = 64$$
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