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 Stefan's law; $P=\sigma Ae{T}^4$
$⟹P=P_o{T}^4$ (where, $P_o=\sigma Ae$)
by taking $log$ at both side;
${\log }_2\left(P/P_o\right)=4{\log }_2\left(T\right)$
$⟹$ For $T_1$,
$1=4{\log }_2\left(760\right)$ ..........(1)
$⟹$ For $T_2$,
Also, ${\log }_2\left(P/P_o\right)=4{\log }_2\left(3040\right)$ ......(2)
Equation (2) - (1), we get ${\log }_2\left(P/P_o\right)-1=4{\log }_2\frac{3040}{760}$
where, ($3040=4\times 760$),
${\log }_2\left(P/P_o\right)-1=4{\log }_2\left(4\right)$ or ${\log }_2\left(P/P_o\right)-1=8$
$⟹{\log }_2\left(P/P_o\right)=9$