A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated by the metal. The sensor has a scale that displays , where is a constant. [2016]

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 $\log_{2} (\frac{P}{P_{0}})$ , where $P_{0}$ is a 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 $2767 ° C$ ? [2016]

Smart Achievers · Accepted Answer

(9)

According to Stefan's law, $P \propto T^{4}$ or $P = P_{0} T^{4}$

$∴$ $\log_{2} P = \log_{2} P_{0} + \log_{2} T^{4}$ $∴$ $\log_{2} \frac{P}{P_{0}} = 4 \log_{2} T$

At $T = 487 ° C = 760 K$ , $\log_{2} \frac{P}{P_{0}} = 4 \log_{2} 760 = 1$ ...(i)

At $T = 2767 ° C = 3040 K$ ,

$\log_{2} \frac{P}{P_{0}} = 4 \log_{2} 3040 = 4 \log_{2} (760 \times 4)$

$= 4 [\log_{2} 760 + \log_{2} 2^{2}] = 4 \log_{2} 760 + 8 = 1 + 8 = 9$

Solution

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Solution

Ans. (9) According to Stefan's law, P∝T4 or P=P0T4 ∴ log2P=log2P0+log2T4 ∴ log2PP0=4log2T At T=487°C=760 K, log2PP0=4log2760=1 ...(i) At T=2767°C=3040 K, log2PP0=4log23040=4log2(760×4) =4[log2760+log222]=4log2760+8=1+8=9

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