A black sphere has radius ' $R$ ' whose rate of radiation is ' $E$ ' at temperature ' $T$ '. if radius is…
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
$3 E$
$16 \mathrm{E}$
E
$9 \mathrm{E}$
Solution
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 $9 \mathrm{E}$
$\mathrm{E}_1=\sigma \times \pi \mathrm{R}_1^2 \times \mathrm{T}_1^4$
$\begin{aligned} & \mathrm{E}_2=\sigma \times \pi \mathrm{R}_2^2 \times \mathrm{T}_2^4 \\ & \therefore \frac{\mathrm{E}_2}{\mathrm{E}_1}=\left(\frac{\mathrm{R}_2}{\mathrm{R}_1}\right)^2\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\right)^4=\left(\frac{1}{3}\right)^2(3)^4=9 \\ & \therefore \mathrm{E}_2=9 \mathrm{E}_1=9 \mathrm{E}\end{aligned}$