Two spheres of radii 4 , m and 1 , m have tem | vedclass

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(A) According to the Stefan-Boltzmann law,the power radiated by a black body is given by $P = \sigma A T^4$. Since the energy emitted $Q$ in time $t$

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$1:1$

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(A) According to the Stefan-Boltzmann law,the power radiated by a black body is given by $P = \sigma A T^4$. Since the energy emitted $Q$ in time $t$ is $Q = P \cdot t = \sigma A t T^4$,the ratio of energy emitted by two spheres is: $\frac{Q_1}{Q_2} = \frac{A_1}{A_2} \left( \frac{T_1}{T_2} ight)^4$. Given $r_1 = 4 \, m$,$r_2 = 1 \, m$,$T_1 = 2000 \, K$,and $T_2 = 4000 \, K$. Since $A = 4 \pi r^2$,the ratio of areas is $\frac{A_1}{A_2} = \left( \frac{r_1}{r_2} ight)^2 = \left( \frac{4}{1} ight)^2 = 16$. The ratio of temperatures is $\frac{T_1}{T_2} = \frac{2000}{4000} = \frac{1}{2}$. Substituting these values: $\frac{Q_1}{Q_2} = 16 	imes \left( \frac{1}{2} ight)^4 = 16 	imes \frac{1}{16} = 1$. Thus,the ratio is $1:1$.

Two spheres of radii $4 \, m$ and $1 \, m$ have temperatures $2000 \, K$ and $4000 \, K$ respectively. Find the ratio of their emitted energy.

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