Calculate Sensor Reading for Thermal Radiation Power at High Temperature

According to Stefan-Boltzmann law, the total thermal radiation power $P$ emitted by a body of emissivity $e$ and surface area $A$ at an absolute temperature $T$ is given by: $P = e \sigma A T^4$

where $\sigma$ is Stefan's constant. Since the emissivity $e$ and area $A$ remain invariant, the radiated power is directly proportional to the fourth power of the absolute temperature: $P \propto T^4$

First, we convert the temperatures from degrees Celsius to Kelvin: $T_1 = 487^\circ\text{C} + 273 = 760\text{ K}$ $T_2 = 2767^\circ\text{C} + 273 = 3040\text{ K}$

The ratio of the final absolute temperature to the initial absolute temperature is: $\frac{T_2}{T_1} = \frac{3040}{760} = 4$

Using the relation $P \propto T^4$, the ratio of the radiated powers is: $\frac{P_2}{P_1} = \left(\frac{T_2}{T_1}\right)^4 = 4^4 = 256 = 2^8$

The sensor readout is defined as $S = \log_2\left(\frac{P}{P_0}\right)$.

For the initial temperature $T_1$, the sensor readout $S_1$ is given as $1$: $S_1 = \log_2\left(\frac{P_1}{P_0}\right) = 1 \implies \frac{P_1}{P_0} = 2^1 = 2$

For the final temperature $T_2$, the new sensor readout $S_2$ is: $S_2 = \log_2\left(\frac{P_2}{P_0}\right) = \log_2\left(\frac{P_2}{P_1} \cdot \frac{P_1}{P_0}\right)$

Substituting the known ratios into the equation: $S_2 = \log_2\left(2^8 \cdot 2^1\right) = \log_2\left(2^9\right) = 9$

Thus, the value shown by the sensor when the temperature is raised to $2767^\circ\text{C}$ is $9$.