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

Answer 1

Given : $T_1={487}^{o}C=487+273=760K$ ; $T_2={2767}^{o}C=2767+273=3040K$

Using $P=\sigma Ae{T}^4$

$⟹P=P_o{T}^4$ where $P_o=\sigma Ae$

OR ${\log }_2\left(P/P_o\right)=4{\log }_2\left(T\right)$

$⟹$ For $T_1=760K$, $1=4{\log }_2\left(760\right)$ ..........(1)

Also ${\log }_2\left(P/P_o\right)=4{\log }_2\left(3040\right)$ ..........(2)

Equation (2) - (1), we get ${\log }_2\left(P/P_o\right)-1=4{\log }_2\frac{3040}{760}$

$\therefore$ ${\log }_2\left(P/P_o\right)-1=4{\log }_2\left(4\right)$

OR ${\log }_2\left(P/P_o\right)-1=8$

$⟹{\log }_2\left(P/P_o\right)=9$