Heat and Thermodynamics 3 Question 29
29. A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated $(P)$ by the metal. The sensor has a scale that displays $\log _2\left(P / P _0\right)$, where $P _0$ is a constant. When the metal surface is at a temperature of $487^{\circ} C$, the sensor shows a value 1 . Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767^{\circ} C$ ?
(2016 Adv.)
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Solution:
- $\quad \log _2 \frac{P _1}{P _0}=1$
Therefore, $\frac{P _1}{P _0}=2$
According to Stefan’s law,
$$ \begin{aligned} P & \propto T^{2} \\ \Rightarrow \quad \frac{P _2}{P _1} & =\left(\frac{T _2}{T _1}\right)^{4}=\left(\frac{2767+273}{487+273}\right)^{4}=4^{4} \\ \Rightarrow \quad \frac{P _2}{P _1} & =\frac{P _2}{2 P _0}=4^{4} \Rightarrow \frac{P _2}{P _0}=2 \times 4^{4} \\ \log _2 \frac{P _2}{P _0} & =\log _2\left[2 \times 4^{4}\right]=\log _2 2+\log _2 4^{4} \\ & =1+\log _2 2^{8}=9 \end{aligned} $$
