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))^(2)`
D. `(T_(2)/T_(1))^(2)`

Answer

C

Explanation

From Stefan's law, `Q/t=sigmaAT^(4)`
i.e. Power = `sigmaAT^(4)`
Since they radiate the same power
`thereforesigmaA_(1)T_(1)^(4)=sigmaA_(2)T_(2)^(4)`
`thereforeA_(2)/A_(1)=T_(1)^(4)/T_(2)^(4)`
`therefore(4pir_(2)^(2))/(4pir_(1)^(2))=(T_(1)/T_(2))^(4)=[(T_(1)/T_(2))^(2)]^(2)`
Taking the square roots, we get
`thereforer_(2)/r_(1)=(T_(1)/T_(2))^(2)" "` ...Option ( c)