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 scale that displays `log_2,(P//P_0)`, where `P_0` is 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`?

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Convert temperatures from Celsius to Kelvin
We need to convert the given temperatures from Celsius to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]

- For \( T_1 = 487 °C \):
\[ T_1 = 487 + 273 = 760 \, K \]

- For \( T_2 = 2767 °C \):
\[ T_2 = 2767 + 273 = 3040 \, K \]

### Step 2: Use the Stefan-Boltzmann Law
The power radiated by a black body is given by the Stefan-Boltzmann Law:
\[ P = \varepsilon \sigma A T^4 \]
Where:
- \( P \) is the power radiated,
- \( \varepsilon \) is the emissivity,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the area,
- \( T \) is the absolute temperature in Kelvin.

Since the emissivity and area are constant, we can express the power radiated at two different temperatures as:
\[ P_1 = P_0 T_1^4 \quad \text{and} \quad P_2 = P_0 T_2^4 \]

### Step 3: Relate the powers using the logarithmic scale
From the problem, we know that:
\[ \log_2\left(\frac{P_1}{P_0}\right) = 1 \]
This implies:
\[ \frac{P_1}{P_0} = 2 \quad \Rightarrow \quad P_1 = 2 P_0 \]

### Step 4: Calculate the ratio of powers at the two temperatures
Using the power relationship:
\[ \frac{P_2}{P_0} = \frac{P_0 T_2^4}{P_0} = T_2^4 \]
Thus, we can express the logarithmic relationship as:
\[ \log_2\left(\frac{P_2}{P_0}\right) = \log_2\left(\frac{P_0 T_2^4}{P_0}\right) = \log_2(T_2^4) = 4 \log_2(T_2) \]

### Step 5: Substitute the values into the logarithmic equation
We know:
\[ \log_2\left(\frac{P_1}{P_0}\right) = 1 \quad \Rightarrow \quad 1 = 4 \log_2(T_1) \]
Substituting \( T_1 = 760 \):
\[ 1 = 4 \log_2(760) \]

Now for \( T_2 = 3040 \):
\[ \log_2\left(\frac{P_2}{P_0}\right) = 4 \log_2(3040) \]

### Step 6: Calculate the new value displayed by the sensor
From the previous steps, we can find:
\[ \log_2\left(\frac{P_2}{P_0}\right) = 4 \log_2(3040) \]
Now we can express the final value displayed by the sensor:
\[ \log_2\left(\frac{P_2}{P_0}\right) = 1 + 4 \log_2\left(\frac{T_2}{T_1}\right) \]
Calculating the ratio:
\[ \frac{T_2}{T_1} = \frac{3040}{760} = 4 \]
Thus:
\[ \log_2(4) = 2 \]
So:
\[ \log_2\left(\frac{P_2}{P_0}\right) = 1 + 4 \times 2 = 1 + 8 = 9 \]

### Final Answer
The value displayed by the sensor when the temperature of the metal surface is raised to \( 2767 °C \) is **9**.

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