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

Answer

To find the temperature of the heat furnace, we can use the Stefan-Boltzmann law of radiation, which states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. The formula we will use is:

\[ E = \varepsilon \sigma A T^4 \]

Where:
- \( E \) = energy radiated per second (in joules)
- \( \varepsilon \) = emissivity of the surface (given as 0.80)
- \( \sigma \) = Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \))
- \( A \) = area of the hole (given as \( 10^{-4} \, \text{m}^2 \))
- \( T \) = absolute temperature in Kelvin

### Step-by-Step Solution:

1. **Convert the energy from calories to joules:**
- Given energy \( E = 1.58 \times 10^5 \) calories/hour.
- We know that \( 1 \, \text{calorie} = 4.184 \, \text{joules} \).
- Therefore, convert calories to joules:
\[ E = 1.58 \times 10^5 \, \text{calories/hour} \times 4.184 \, \text{joules/calorie} \]
- Convert hours to seconds:
\[ E = \frac{(1.58 \times 10^5 \times 4.184)}{3600} \, \text{joules/second} \]

2. **Calculate the energy in joules per second:**
- Calculate \( E \):
\[ E \approx \frac{(1.58 \times 10^5 \times 4.184)}{3600} \approx 184.5 \, \text{joules/second} \]

3. **Substitute values into the Stefan-Boltzmann equation:**
- Now, substitute \( E \), \( \varepsilon \), \( \sigma \), and \( A \) into the equation:
\[ 184.5 = 0.80 \times 5.67 \times 10^{-8} \times 10^{-4} \times T^4 \]

4. **Rearranging the equation to solve for \( T^4 \):**
- Rearranging gives:
\[ T^4 = \frac{184.5}{0.80 \times 5.67 \times 10^{-8} \times 10^{-4}} \]

5. **Calculate \( T^4 \):**
- Calculate the denominator:
\[ 0.80 \times 5.67 \times 10^{-8} \times 10^{-4} = 4.536 \times 10^{-12} \]
- Now calculate \( T^4 \):
\[ T^4 = \frac{184.5}{4.536 \times 10^{-12}} \approx 4.07 \times 10^{13} \]

6. **Calculate \( T \):**
- 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 \( 2500 \, \text{K} \).