The intensity of salr radiation just outside the earth's atmosphere is measured to be `1.4 kW//m^2`. If the radius of the sun `7xx10^8m`, while the earth-sun distance is `150xx10^6km`, then find
i. the intensity of salr radiation at the surface of the sun.
ii. the temperature at the surface of the sun assuming it to be a black body,
iii. the most probable wavelength in solar radiation.

Explanation

Assuming the sun to be a blackboy at a temperature `T_0`, we can write,
`W=`intensity of solar radiation on the sun's surface`=sigmaT_0^4`, Where `sigma` is the stefan-Boltzmann constant.
i. The radiation emitted from the solar surface per unit time is spread over the surface of a sphere having a radius equal to earth-sun distance where it is received on the earth (just outside the atmosphere)
`Wxx4piR_S^2=I_0xx4piD_(SE)^2`
Where `D_(SE)` is the distance between the sun and the earth, and `I_0` is the intensity outside the earth's atmosphere
`I_0=Wxx((R_S)/(D_(SE)^(2)))`
Now, `R_S=7xx10^8m`,`D_(SE)=150xx10^9m`
and `I_0=1.4xx10^3 W//m^2`
`:. 1.4xx10^3=Wxx((7xx10^8)/(150xx10^9))^2=Wxx(49)/(225)xx10^-4`
or `W=6.4xx10^7(W)/(m^2)`
ii. Assuming the sun to be a black body,
`6.4xx10^4=sigmaT_0^4=(5.67xx10^-8)T_0^4`
`:. T_0^4=(6.4)/(5.67)xx10^(15)`
or `T_0approx0.58xx10^4K=5800K`
iii. Using Wien's Displacement law
`lamda_("mp")T_0=0.29cm-K=2.9xx10^-3m-K`
or `lamda_("mp")=(2.0xx10^-3)/(5800)=5xx10^-7m=5000Å`