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  4. The radiation emitted by a star A is 10000 times that of the sun. If the surface temperatures of the sun and star A are 6000 K and 2000 K respectively, the ratio of the radii of the star A and the sun is

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Q. The radiation emitted by a star $A$ is $10000$ times that of the sun. If the surface temperatures of the sun and star $A$ are $6000 \, K$ and $2000 \, K$ respectively, the ratio of the radii of the star $A$ and the sun is

NTA AbhyasNTA Abhyas 2020Thermal Properties of Matter

Solution:

$\epsilon \propto AT^{4} \, ; \, \epsilon \propto R^{2}T^{4}$ Or $R \propto \sqrt{\frac{\epsilon }{T^{4}}}$
$\frac{R_2}{R_1}=\sqrt{\frac{\varepsilon_2}{\varepsilon_1} \cdot \frac{T_1^4}{T_2^4}}=\sqrt{\frac{10000}{1} \times\left(\frac{6000}{2000}\right)^4}$
$=\left(10\right)^{2}\times \left(\frac{3}{1}\right)^{2}=900:1$