The area of a hole of heat furnace is `10^(-4)m^(2)`. It radiates `1.58xx10^(5)` calories of heat per hour. If the emissivity of the furnace is 0.80, then its temperature is

Options

A. 1500 K
B. 2000 K
C. 2500 K
D. 3000 K

Explanation

To find the temperature of the heat furnace using the given data, we will apply the Stefan-Boltzmann Law. Here are the steps to solve the problem:

### Step 1: Understand the Stefan-Boltzmann Law
The Stefan-Boltzmann Law states that the power radiated by a black body per unit area is proportional to the fourth power of its absolute temperature. The formula is given by:
\[ P = e \cdot \sigma \cdot A \cdot T^4 \]
where:
- \( P \) = Power (in watts)
- \( e \) = Emissivity (dimensionless)
- \( \sigma \) = Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \))
- \( A \) = Area (in m²)
- \( T \) = Temperature (in Kelvin)

### Step 2: Convert the Given Power from Calories to Watts
The power \( P \) is given as \( 1.58 \times 10^5 \) calories per hour. We need to convert this to watts (joules per second):
1 calorie = 4.184 joules
1 hour = 3600 seconds

Thus,
\[ P = 1.58 \times 10^5 \, \text{cal/hr} \times \frac{4.184 \, \text{J}}{1 \, \text{cal}} \times \frac{1 \, \text{hr}}{3600 \, \text{s}} \]
Calculating this gives:
\[ P = 1.58 \times 10^5 \times 4.184 \div 3600 \approx 184.5 \, \text{W} \]

### Step 3: Substitute the Known Values into the Stefan-Boltzmann Equation
Now we can substitute the known values into the Stefan-Boltzmann equation:
\[ 184.5 = 0.80 \cdot (5.67 \times 10^{-8}) \cdot (10^{-4}) \cdot T^4 \]

### Step 4: Simplify the Equation
First, calculate the product of the constants:
\[ 0.80 \cdot (5.67 \times 10^{-8}) \cdot (10^{-4}) = 4.536 \times 10^{-12} \]
Now, substituting this back into the equation gives:
\[ 184.5 = 4.536 \times 10^{-12} \cdot T^4 \]

### Step 5: Solve for \( T^4 \)
Rearranging the equation to solve for \( T^4 \):
\[ T^4 = \frac{184.5}{4.536 \times 10^{-12}} \approx 4.07 \times 10^{13} \]

### Step 6: Calculate \( T \)
Now take the fourth root to find \( T \):
\[ T = (4.07 \times 10^{13})^{1/4} \approx 2500 \, \text{K} \]

### Final Answer
The temperature of the furnace is approximately \( 2500 \, \text{K} \).

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