Question 1: Thermodynamics & KTG

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 ${\log _2}\left( {{P \over {{P_0}}}} \right)$, 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?

Accepted Answer

${\log _2} = {{{p_1}} \over {{p_0}}} = 1$
Therefore, ${{{p_1}} \over {{p_0}}} = 2$
According to Stefan's law, p $ \propto $ T
2
$ \Rightarrow {{{p_2}} \over {{p_1}}} = {\left( {{{{T_2}} \over {{T_1}}}} ight)^4} = {\left( {{{2767 + 273} \over {487 + 273}}} ight)^4} = {4^4}$
${{{p_2}} \over {{p_1}}} = {{{p_2}} \over {2{p_0}}} = {4^4} \Rightarrow {{{p_2}} \over {{p_0}}} = 2 	imes {4^4}$
${\log _2}{{{p_2}} \over {{p_0}}} = {\log _2}[2 	imes {4^4}]$
$ = {\log _2}2 + {\log _2}{4^4}$
$ = 1 + {\log _2}{2^8} = 1 + 8 = 9$

Thermodynamics & KTG

Integer / Numeric Answer Type