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(A) According to the Stefan-Boltzmann Law,the energy radiated per unit area per unit time $(E)$ is proportional to the fourth power of the absolute te

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$5000 \ K$

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(A) According to the Stefan-Boltzmann Law,the energy radiated per unit area per unit time $(E)$ is proportional to the fourth power of the absolute temperature $(T)$: $E \propto T^4$.Given:$E_1 = 1 	imes 10^5 \ J/s \cdot m^2$$T_1 = 227^\circ C = 227 + 273 = 500 \ K$$E_2 = 1 	imes 10^9 \ J/s \cdot m^2$Using the ratio formula:$\frac{E_2}{E_1} = \left( \frac{T_2}{T_1} ight)^4$Substituting the values:$\frac{1 	imes 10^9}{1 	imes 10^5} = \left( \frac{T_2}{500} ight)^4$$10^4 = \left( \frac{T_2}{500} ight)^4$Taking the fourth root on both sides:$10 = \frac{T_2}{500}$$T_2 = 10 	imes 500 = 5000 \ K$

$A$ black body radiates energy at the rate of $1 \times 10^5 \ J/s \cdot m^2$ at a temperature of $227^\circ C$. The temperature to which it must be heated so that it radiates energy at a rate of $1 \times 10^9 \ J/s \cdot m^2$ is:

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