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(D) The power radiated by a black body is given by the Stefan-Boltzmann law: $P = A \sigma T^4 = 4 \pi r^2 \sigma T^4$.From this relation,we can see t

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$1800$

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(D) The power radiated by a black body is given by the Stefan-Boltzmann law: $P = A \sigma T^4 = 4 \pi r^2 \sigma T^4$.From this relation,we can see that $P \propto r^2 T^4$.Given the initial conditions: $r_1 = 12 \, cm$,$T_1 = 500 \, K$,and $P_1 = 450 \, W$.New conditions: $r_2 = \frac{r_1}{2}$ and $T_2 = 2 T_1$.The ratio of the powers is given by: $\frac{P_2}{P_1} = \left( \frac{r_2}{r_1} ight)^2 \left( \frac{T_2}{T_1} ight)^4$.Substituting the values: $\frac{P_2}{P_1} = \left( \frac{1}{2} ight)^2 	imes (2)^4 = \frac{1}{4} 	imes 16 = 4$.Therefore,$P_2 = 4 	imes P_1 = 4 	imes 450 \, W = 1800 \, W$.

$A$ spherical black body of radius $12 \, cm$ radiates $450 \, W$ of power at $500 \, K$. If the radius is halved and the temperature is doubled,the power radiated will be ........ $W$.

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