Two spherical black bodies have radii R_1 and R_2 . Their surface...

According to the Stefan-Boltzmann law, the power radiated by a black body is given by $P = \sigma A T^4$.

For a spherical body of radius $R$, the surface area is $A = 4 \pi R^2$.

The radiated power is $P = \sigma (4 \pi R^2) T^4$.

Given that both bodies radiate the same power, $P_1 = P_2$.

$\sigma (4 \pi R_1^2) T_1^4 = \sigma (4 \pi R_2^2) T_2^4$

$R_1^2 T_1^4 = R_2^2 T_2^4$

$\dfrac{R_1^2}{R_2^2} = \dfrac{T_2^4}{T_1^4}$

$\dfrac{R_1}{R_2} = \left(\dfrac{T_2}{T_1}\right)^2$

Answer: \left(\dfrac{T_2}{T_1}\right)^2