Question

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated $\left(P\right)$ by the metal. The sensor has a scale that displays ${\text{log}}_2\left(P/P_0\right)$, where $P_0$ is a constant. When the metal surface is at a temperature of ${487}^{o}C$, 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 ${2767}^{o}C$?

Answer 1

Let display is $X={\log }_2\left(\frac{P}{P_0}\right)$

At temperature $T_1={487}^{o}C=760K$, Power is $P_1$

$X_1={\log }_2\left(\frac{P_1}{P_0}\right)$

At $T_2={2767}^{o}C=3040K$, Power is $P_2$

$X_2={\log }_2\left(\frac{P_2}{P_0}\right)$

Now, $X_2{X}_1={\log }_2\left(\frac{P_2}{P_1}\right)$

From Stefan’s law, $\frac{P_2}{P_1}={\left(\frac{T_2}{T_1}\right)}^4$

$X_2{X}_1=4{\log }_2\left(\frac{T_2}{T_1}\right)=4{\log }_{2}4=8$

$X_2=8+X_1=9$