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Compare the rate of loss of heat from a metal sphere at $627^{\circ} \mathrm{C}$ with the rate of loss of heat from the same sphere at $327^{\circ} \mathrm{C}$, if the temperature of the surrounding is $27^{\circ} \mathrm{C}$. (nearly)
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Verified Answer
The correct answer is:
$5.3$
The heat loss is given as $\mathrm{R}=\mathrm{e} \sigma \mathrm{A}\left(\mathrm{T}^4-\mathrm{T}_0^4\right)$
Let surrounding temperature be denoted as $\mathrm{T}$ Heat loss from the metal sphere at a temperature $\mathrm{T}_1$
$$
\mathrm{R}_1=\operatorname{e} \sigma \mathrm{A}\left(\mathrm{T}_1^4-\mathrm{T}_0^4\right)
$$
Heat loss from the metal sphere at a temperature $\mathrm{T}_2$
$$
R_2=e \sigma A\left(T_2^4-T^4\right)
$$
$$
\begin{aligned}
\therefore \quad \frac{\mathrm{R}_1}{\mathrm{R}_2} & =\frac{\mathrm{T}_1^4-\mathrm{T}^4}{\mathrm{~T}_2^4-\mathrm{T}^4}=\frac{900^4-300^4}{600^4-300^4} \\
\frac{\mathrm{R}_1}{\mathrm{R}_2} & =5.3
\end{aligned}
$$
Let surrounding temperature be denoted as $\mathrm{T}$ Heat loss from the metal sphere at a temperature $\mathrm{T}_1$
$$
\mathrm{R}_1=\operatorname{e} \sigma \mathrm{A}\left(\mathrm{T}_1^4-\mathrm{T}_0^4\right)
$$
Heat loss from the metal sphere at a temperature $\mathrm{T}_2$
$$
R_2=e \sigma A\left(T_2^4-T^4\right)
$$
$$
\begin{aligned}
\therefore \quad \frac{\mathrm{R}_1}{\mathrm{R}_2} & =\frac{\mathrm{T}_1^4-\mathrm{T}^4}{\mathrm{~T}_2^4-\mathrm{T}^4}=\frac{900^4-300^4}{600^4-300^4} \\
\frac{\mathrm{R}_1}{\mathrm{R}_2} & =5.3
\end{aligned}
$$
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