Two spherical black bodies of radii `R_(1)` and `R_(2)` and with surface temperature `T_(1)` and `T_(2)` respectively radiate the same power. `R_(1)//R_(2)` must be equal to

Options

A. `(T_(1))/(T_(2))`
B. `(T_(2))/(T_(1))`
C. `((T_(1))/(T_(2)))^(3)`
D. `((T_(2))/(T_(1)))^(2)`

Answer

C

Explanation

The rate at which an object radiates energy is given as,
`Q/t sigma AT^(4) rArr` Power, `P = sigmaAT^(4)`
`:. A prop 1/(T^(4))`
It the body radiates same power, then
`rArr (A_(2))/(A_(1)) = (T_(1)^(2))/(T_(2)^(4)) rArr (4pir_(2)^(2))/(4pir_(1)^(2)) = (T_(1)^(4))/(T_(2)^(4))`
`:. (r_(2))/(r_(1)) = ((T_(1))/(T_(2)))^(2)`