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A black sphere has radius ' $R$ ' whose rate of radiation is ' $\mathrm{E}$ ' at temperature ' $\mathrm{T}$ '. If radius is made $R / 3$ and temperature ' $3 \mathrm{~T}$ ', the rate of radiation will be
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$9 \mathrm{E}$
$\begin{array}{ll} & \mathrm{E}=\mathrm{eA} \sigma \mathrm{T}^4 \\ & \text { But for sphere, } A=4 \pi \mathrm{R}^2 \\ \therefore \quad & \mathrm{E}=\mathrm{e}\left(4 \pi \mathrm{R}^2\right) \sigma \mathrm{T}^4 \\ \therefore \quad & \frac{\mathrm{E}_1}{\mathrm{E}_2}=\frac{\mathrm{R}_1^2 \mathrm{~T}_1^4}{\mathrm{R}_2^2 \mathrm{~T}_2^4} \\ \therefore \quad & \frac{\mathrm{E}}{\mathrm{E}_2}=\frac{\mathrm{R}^2 \mathrm{~T}^4}{\left(\frac{\mathrm{R}}{3}\right)^2(3 \mathrm{~T})^4} \\ \therefore \quad & \mathrm{E}_2=9 \mathrm{E}\end{array}$
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