A spherical black body with a radius of $12\ cm$ radiates $450\ watt$ power at $500\ K$. If the radius were halved and the temperature doubled, the power radiated in watt would be
NEET 2017, Medium
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According to $Stefan-Boltzman$ law, rate of energy radiated by a black is given as
$E = \sigma A{T^4} = \sigma 4\pi {R^2}{T^4}$
$Given\,{E_1} = 450\,W,\,{T_1} = 500\,K,{R_1} = 12\,cm$
${R_2} = \frac{{{R_1}}}{2},{T_2} = 2{T_1},{E_2} = ?$
$\frac{{{E_2}}}{{{E_1}}} = \frac{{\sigma 4\pi R_2^4T_2^4}}{{\sigma 4\pi R_1^2T_1^4}} = {\left( {\frac{{{R_2}}}{{{R_1}}}} \right)^2}{\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^4}$
$\frac{{{E_2}}}{{{E_1}}} = \frac{1}{4} \times 16 = 4$
${E_2} = {E_1} \times 4 = 450 \times 4 = 1800\,W$
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