(Temperature of Sun)
Esimate the temperature `T_(c )` of the earth, assuming it is in radiative equilibrium with the sun. (Assume radius of sun, `r_(s)=7xx10^(8)m`, temperature of solar surface`=6000K`, distance of earth from sun,`R=1.5xx10^(11)m`)

Explanation

Power radiated from sun `=sigmaxx`surface area `xxT^(4)`
`=sigmaxx4pir_(s)^(2)xxT_(s)^(4)`
Power received by earth `=(pir_(e)^(2))/(4piR^(2))xx` power radiated by sun
since `pir_(e)^(2)` is the effective area of the earth on which the sun's radiation is incident normally. figure and `4piR^(2)` is the total area on which the sun's radiation falls at a distance `R` from the sun where the earth is situated.
Now power radiated by earth `=sigma*4pir_(e)^(2)*T_(e)^(4)`

Assuming radiative equilibrium
power radiated by earth `=` power received by earth
`:. sigma*4pir_(e)^(2)*T_(e)^(2)=sigma*4pir_(s)^(2)*T_(s)^(4)xx(pir_(c )^(2))/(4piR^(2))`
Cancelling `r_(e)^(2)` and simplifying, then
`T_(e)^(4)=T_(s)^(4)xx((r_(s)^(2))/(4R^(2)))`
`:. T_(e)=T_(s)xx((7xx10^(8))/(2xx1.5xx10^(11)))^(1//2)=209K`
Note that the calculation is approximate, for example, the earth and the sun are not perfect black body radiators and the earth receives heat from its interior.