The radiation emitted by a star $A$ is 10000 times that of the sun. If the surface temperatures of the sun…
The radiation emitted by a star $A$ is 10000 times that of the sun. If the surface temperatures of the sun and the star $A$ are $6000 \mathrm{~K}$ and $2000 \mathrm{~K}$ respectively, the ratio of the radii of the star $A$ and the sun is :
$300: 1$
$600: 1$
$900: 1$
$1200: 1$
Solution
Energy radiated per unit time
$E=\sigma A T^4$
where $\sigma=$ Stefan's constant
$\therefore \quad$ For sun $E_{\text {sun }}=\sigma A_{\text {sun }} T_{\text {sun }}^4$
According to question,
$E_{\text {star }}=10000 E_{\text {sun }}$
$\sigma A_{\text {star }} \times T_{\text {star }}^4=10000 \times \sigma A_{\text {sun }} \times T_{\text {sun }}^4$
$\pi R_{\text {star }}^2 T_{\text {star }}^4=10000 \times \pi R_n^2 \times T_{\text {sun }}^4$
$\left(\frac{R_{\text {star }}}{R_{\text {sun }}}\right)^2=10000\left(\frac{T_{\text {sun }}}{T_{\text {star }}}\right)^4$
$=10000\left(\frac{6000}{2000}\right)^4$
$\Rightarrow \quad \frac{R_{\text {star }}}{R_{\text {sun }}}=\sqrt{10000 \times(3)^4}$
$=100 \times 3^2=900$
$R_{\text {star }}: R_{\text {sun }}=900: 1$