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(B) The power $P$ received by a small area at a distance $d$ from a spherical source at temperature $T$ is proportional to the Stefan-Boltzmann law an

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$4 P$

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(B) The power $P$ received by a small area at a distance $d$ from a spherical source at temperature $T$ is proportional to the Stefan-Boltzmann law and the inverse square law of radiation.According to the Stefan-Boltzmann law,the total power radiated by the sphere is $P_{source} \propto T^4$.The intensity of radiation at a distance $d$ follows the inverse square law,$I \propto \frac{1}{d^2}$.Thus,the power received by the foil is $P \propto \frac{T^4}{d^2}$.Let the initial state be $P_1 = k \frac{T^4}{d^2} = P$.In the final state,the temperature becomes $T' = 2T$ and the distance becomes $d' = 2d$.The new power $P'$ is given by:$P' = k \frac{(2T)^4}{(2d)^2} = k \frac{16 T^4}{4 d^2} = 4 \left( k \frac{T^4}{d^2} \right) = 4P$.

$A$ black metal foil is warmed by radiation from a small sphere at temperature $T$ and at a distance $d$. It is found that the power received by the foil is $P$. If both the temperature and distance are doubled,the power received by the foil will be:

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