Question

The surface of the earth receives solar radiation at the rate of 1400 $W{m}^{-2}$. The distance of the centre of the sun from the surface of the earth is $1.5\times {10}^{11}m$ and the radius of the sun is $7.0\times {10}^{8}m$. Treating the sun as a back body, it follows from the above data that the surface temperature of the sun is about

• A. 5800 K
• B. 5900 K
• C. 6000 K
• D. 6100 K

Answer 1

The correct option is A

5800 K

Let R be the radius of the sun and let r be the radius of the earth’s orbit around the sun. If the surface temperature of the sun is T (in kelvin), the energy emitted per second by the surface of the sun $=4\pi R^2\sigma T^4,where\sigma T^4,where\sigma$ is Stefan’s constant whose value is $5.67\times {10}^{-8}W{m}^{-2}{K}^{-4}$. Now the area of the spherical surface of radius r is $4\pi r^2$. Therefore, energy received per second by a unit area of the earth’s surface is

$\frac{4\pi R^2\sigma T^4}{4\pi r^2}=\frac{R^2\sigma T^4}{r^2}=1400$

Which gives

$T^4=\frac{1400r^2}{\sigma R^2}=\frac{1400\times (1.5\times {10}^{11}{)}^2}{(5.67\times {10}^{-8})\times (7\times {10}^8{)}^2}=1.1338\times {10}^{15}T=5803K$

Hence the correct choice is (a).