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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(P/P_{0}) where P₀ is a constant. When the metal surface is at a temperature of 487⁰C, 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 2767⁰C?

Answer:

Explanation:

  $\log_{2}\frac{P_{1}}{P_{0}}=1$

Therefore $\frac{P_{1}}{P_{2}}=2$

According Stefan's law $p\proptoT$

  $\frac{P_{2}}{P_{1}}=\left(\frac{T_{2}}{T_{1}}\right)^{4}=\left(\frac{2767+273}{487+273}\right)^{4}=4^{4}$

$\frac{P_{2}}{P_{1}}=\frac{P_{2}}{2P_{0}}=4^{4}$

$\Rightarrow \frac{P_{2}}{P_{0}}=2\times 4^{4}$

$\log_{2}\frac{P_{2}}{P_{0}}=\log_{2}[2\times 4^{4}]$

$=\log_{2}2+\log_{2}4^{4}$

$1+\log_{2}2^{8}=1+8=9$