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 scale that displays `log_2,(P//P_0)`, where `P_0` is constant. When the metal surface is at a temperature of `487^@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^@C`?

Answer

9

Explanation

Given at temperature
`T_(1) =487^(@)C =487 + 273 = 760 K`
If R is the power radiated at this temperature, then
`log_(2)((P_(1))/(P_(0)))=1` or `(P_(1))/(P_(0))=2` or `P_(1) =2 P_(0)`
At temperature `T_(2) =2767^(@)C =2767 + 273 = 3040 K`
Let `P_(2)` be the power radiated at this temperature. Ac cording to Stefan's law
`P prop T^(4)`
`:. (P_(2))/(P_(1)) =((T_(2))/(T_(1)))^(4) = ((3040)/(760))^(4) = 4^(4)`
`(P_(2))/(P_(1)) = (P_(2))/(2P_(0)) =4^(4)` or `(P_(2))/(P_(0)) =2xx4^(4`
`log_(2) ((P_(2))/(P_(0))) =log_(2)(2xx4^(4)) = log_(2) 2 + log_(2) 4^(4)`
`=1 + log_(2) 2^(8) =1 + 8 =9`