Question

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 $lo{g}_2\left(P/P_0\right)$, 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 ${2767}^{\circ }C?$

Answer 1

Given: $lo{g}_2\left(\frac{P_1}{P_0}\right)=1\Rightarrow P_0=\frac{P_1}{2}$
According to Stefan's law,
$\frac{P_2}{P_1}={\left(\frac{T_2}{T_1}\right)}^4=\frac{(2767+273{)}^4}{(487+273{)}^4}=4^4$
$\frac{P_2}{P_1}=\frac{P_2}{2P_0}=4^4\Rightarrow \frac{P_2}{P_0}=2\times 4^4=4^{9/2}$.
$\therefore lo{g}_2\left(\frac{P_2}{P_0}\right)=lo{g}_2(4{)}^{9/2}=lo{g}_2{2}^9=9.$