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

Solution :
$$\eqalign{ & E = \sigma A{T^4};\,\,A\,\propto \,{R^2} \cr & \therefore \,\,E\,\propto \,{R^2}{T^4} \cr & \therefore \,\,\frac{{{E_2}}}{{{E_1}}} = \frac{{R_2^2\,T_2^4}}{{R_1^2T_1^4}} \cr & \Rightarrow \,\,\frac{{{E_2}}}{{{E_1}}} = \frac{{{{\left( {2\,R} \right)}^2}{{\left( {2\,T} \right)}^4}}}{{{R^2}{T^4}}} \cr & = 64 \cr} $$