A black sphere has radius ' \(R\) ' whose rate of radiation is ' \(E\) ' at temperature ' \(T\) '. if radius is made ' \(\frac{R}{3}\) 'and temperature ' \(3 T\) ', the rate of radiation will be

A black sphere has radius ' \(\mathrm{R}\) ' whose rate of radiation is ' \(\mathrm{E}\) ' at temperature ' \(\mathrm{E}\) ' ' \(\mathrm{T}\) '. If radius is made \(\frac{\mathrm{R}}{3}\) and temperature ' \(3 \mathrm{~T}\) ', the rate of radiation will be \(9 \mathrm{E}\)
\(\mathrm{E}_1=\sigma \times \pi \mathrm{R}_1^2 \times \mathrm{T}_1^4\)
\(\begin{aligned} & \mathrm{E}_2=\sigma \times \pi \mathrm{R}_2^2 \times \mathrm{T}_2^4 \\ & \therefore \frac{\mathrm{E}_2}{\mathrm{E}_1}=\left(\frac{\mathrm{R}_2}{\mathrm{R}_1}\right)^2\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\right)^4=\left(\frac{1}{3}\right)^2(3)^4=9 \\ & \therefore \mathrm{E}_2=9 \mathrm{E}_1=9 \mathrm{E}\end{aligned}\)