Two spherical black bodies of radii $\Gamma_1$ and $\Gamma_{2}$ and with surface temperature $T_1$ and $T_2$ respectively radiate the same power. Then the ratio of $\Gamma_1$ and $\Gamma_{2}$ will be
$(a) \left(\frac{T_2}{T_1}\right)^2 \quad (b) \left(\frac{T_2}{T_1}\right)^4 (c) \left(\frac{T_1}{T_2}\right)^2 \quad (d) \left(\frac{T_1}{T_2}\right)^4$
Text Solution
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$For black body, P = AεσT^{4}. For same power A \propto \frac{1}{T^{4}} \Rightarrow \left(\frac{r_{1}}{r_{2}}\right)^{2} = \left(\frac{T_{2}}{T_{1}}\right)^{4} \Rightarrow \frac{r_{1}}{r_{2}} = \left(\frac{T_{2}}{T_{1}}\right)^{2}$
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