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(A) According to the Stefan-Boltzmann law,the total power $P$ radiated by a sphere of radius $r$ and temperature $T$ is given by $P = A \sigma T^4$,wh

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$(\frac{T_2}{T_1})^2$

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(A) According to the Stefan-Boltzmann law,the total power $P$ radiated by a sphere of radius $r$ and temperature $T$ is given by $P = A \sigma T^4$,where $A = 4 \pi r^2$ is the surface area and $\sigma$ is the Stefan-Boltzmann constant.Since the material is the same,the emissivity is the same. Given that the total power radiated $P$ is the same for both spheres,we have $P_1 = P_2$.Substituting the formula: $4 \pi r_1^2 \sigma T_1^4 = 4 \pi r_2^2 \sigma T_2^4$.Simplifying the equation: $r_1^2 T_1^4 = r_2^2 T_2^4$.Rearranging to find the ratio of radii: $(\frac{r_1}{r_2})^2 = (\frac{T_2}{T_1})^4$.Taking the square root on both sides: $\frac{r_1}{r_2} = (\frac{T_2}{T_1})^2$.

Two spheres of the same material have radii $r_1$ and $r_2$ and surface temperatures $T_1$ and $T_2$ respectively. If their emissive power (total power radiated) is the same,find the ratio of their radii $r_1/r_2$.

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