The sun having surface temperature `Ts` radiates like a black body. The rasius of sun is `Rs` and earth is at a distance R from the surface of sun. Earth absorbs radiations falling on its surface from sun only and is at constant temperature T. If radiations falling on earth's surface are alomost parallel and earth also radiates like a blackbody, then

Options

A. `T = T_(s) sqrt(R_(s)/(2R))`
B. `T = Ts`
C. `T = (T_(s))/(2) sqrt((R_(s)/(R ))`
D. `T = T_(s) sqrt((R_(s)/(R ))`

Answer

A

Explanation

Energy absorbed per unit time by earth's surface, if its radius is r, is
`Q_(abs) = (sigma4piR_(s)^(2)T_(s)^(4))/(4piR^(2))xxpir^(2)`
Energy radiated per unit time by earth's surface, `Q_(rad) = sigma(4pir^(2))T^(4)`
In equilibrium
`Q_(abs)=Q_(rad)`
`rArr T^(4)=(T_(s)^(4))/(4)(R_(s)^(2))/(R^(2))`