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 $\log_{2}\,(P/P_0)$, where $P_0$ is a constant. When the metal surface is at a temperature of $487^\circ\text{C}$, 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 $2767^\circ\text{C}$?

Step 1: Convert temperatures to Kelvin.

Convert the given Celsius temperatures to absolute temperatures.

$$T_1 = 487 + 273 = 760\text{ K}$$ $$T_2 = 2767 + 273 = 3040\text{ K}$$

Calculate the ratio of the temperatures.

$$\frac{T_2}{T_1} = \frac{3040}{760} = 4$$

Step 2: Relate radiated power to temperature.

According to the Stefan-Boltzmann Law, the power radiated by a body is proportional to the fourth power of its absolute temperature.

$$P = e\sigma A T^{4}$$

Since emissivity $e$, Stefan-Boltzmann constant $\sigma$, and surface area $A$ are constant:

$$P \propto T^{4}$$

Step 3: Determine the ratio of the radiated powers.

Use the temperature ratio to find the power ratio.

$$\frac{P_2}{P_1} = \left(\frac{T_2}{T_1}\right)^{4}$$ $$\frac{P_2}{P_1} = 4^{4}$$ $$\frac{P_2}{P_1} = 256$$ $$\frac{P_2}{P_1} = 2^{8}$$

Step 4: Calculate the new sensor reading.

The sensor reading $r$ is defined by the logarithmic scale.

$$r = \log_{2}\left(\frac{P}{P_0}\right)$$

For the initial temperature, the reading is 1.

$$r_1 = \log_{2}\left(\frac{P_1}{P_0}\right) = 1$$

Express the new reading $r_2$ in terms of the initial reading and the power ratio.

$$r_2 = \log_{2}\left(\frac{P_2}{P_0}\right)$$ $$r_2 = \log_{2}\left(\frac{P_2}{P_1} \cdot \frac{P_1}{P_0}\right)$$ $$r_2 = \log_{2}\left(\frac{P_2}{P_1}\right) + \log_{2}\left(\frac{P_1}{P_0}\right)$$

Substitute the known values into the equation.

$$r_2 = \log_{2}(2^{8}) + 1$$ $$r_2 = 8 + 1$$ $$r_2 = 9$$

Answer: 9