JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Heat & Thermodynamics

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

lo g_{2} (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?

Answer: 9

Solution

By Stefan-Boltzmann law, power radiated

P = e σ A T^{4}

.

At 487°C = 760 K:

P_{1} = e σ A (760)^{4}

Given

lo g_{2} (P_{1} / P_{0}) = 1

, so

P_{1} = 2 P_{0}

, thus

P_{0} = P_{1} /2

.

At 2767°C = 3040 K:

P_{2} = e σ A (3040)^{4}

The ratio:

\frac{P _{2}}{P _{1}} = (\frac{3040}{760})^{4} = 4^{4} = 256

Therefore:

\frac{P _{2}}{P _{0}} = \frac{P _{2}}{P _{1} /2} = 2 \times 256 = 512 = 2^{9}

Sensor reading:

lo g_{2} (P_{2} / P_{0}) = lo g_{2} (2^{9}) = 9

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Integer answer

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JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Heat & Thermodynamics

Solution

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