Question

The earth receives on its surface radiation from the sun at the rate of $1400W{m}^{-2}$. The distance of the centre of the sun from the surface of the earth is $1.5\times {10}^{11}$, and the radius of the sun is $7\times {10}^{8}m$. Treating the sun as a black body, it follows from the above data that its surface temperature is _______ $K$.

Answer 1

$Given$ :- Rate of radiation $=1400W{m}^{-2}$

Distance between sun's centre and earth's surface $\left(R_{es}\right)=1.5\times {10}^{11}m$

Radius of the sun (R_s) $=7\times {10}^{8}m$

$ToFind$ :- Sun's surface Temperature $\left(T\right)$ in $K$

$Solution$ :- We know ,

$EnergyEmitted=Epsilon\times Sigma\times T^4$

Since considering Sun as a black body

$\therefore$ $Epsilon=1$

$⟹$ $E=\sigma T^4$

where $\sigma$ ( Stefan Boltzman Constant )$=5.67\times {10}^{-8}$

Now , Total energy radiated per second by the sun per unit solid angle at the surface of the earth is

$E=\frac{\sigma T^4\times (R_s{)}^2}{(R_{es}{)}^2}$

$1400=\frac{(5.67\times {10}^{-8})T^4\times (7.0\times {10}^8{)}^2}{(1.5\times {10}^{11}{)}^2}$

$T=5801K$

Hence , $\underset{–}{\text{Sun's surface temperature is T =5801 K}}$