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_(2))/(T_(1)))^(2)`
B. `((T_(2))/(T_(1)))^(4)`
C. `((T_(1))/(T_(2)))^(2)`
D. `((T_(1))/(T_(2)))^(4)`

Answer

A

Explanation

Radius of Ist body = `r_(1)` radius of Iind body = `r^_(2)`
Temperature of Ist body = `T_(1)`
Temperature of Iind body = `T_(2)`
The emissivity of a black body or radiated power is given
`"by, "E=AsigmaT^(4)xxtpropAT^(4)`
or `Aprop(E)/(T^(4))` (where,A is the surface area of the spherical black body)
As in the condition of question , the power radiated by Ist and IInd body is same.
`"Hence " Aprop(1)/(T^(4))`
So , `(A_(1))/(A_(2))=((T_(2))/(T_(1)))^(4)rArr(4pir_(1)^(2))/(4pir_(2)^(2))=((T_(2))/(T_1))^(4)`
`"Thus "(r_(1))/(r_(2))=((T_(2))/(T_(1)))^(2)`