Question

The earth receives at its surface radiation from the sun at the rate of the $1400W/m^2$. 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 black body, it flows from the above data that its surface temperature is

Answer 1

Energy recived by earth per $m^2$ per sec $=1400J$

$E_{totoal}\Rightarrow$ Energy recived by the total sphere (imaginary) with center at sun and radius equal to earth -sun distance $(7\times {10}^{8}m)=1400\times [4\pi \times (7\times {10}^8\left)\right]$

$\Rightarrow E_{total}=1400\times 4\pi \times 49\times {10}^{16}J$

$=$ total energy radiated by sun per second

we know that, Energy radiation $=\sigma T^4$

$1400\times 4\pi \times 49\times {10}^{16}=2.898\times {10}^{-3}\times T^4$

$\Rightarrow T^4=297.464\times {10}^{19}$

$\Rightarrow T=23.35K$