The radiant power of a furnace of surface area of `0.6m^(2)` is `34.2KW` The temperature of the furance s `[sigma = 5.7 xx 10^(-8) Wm^(-2) K^(-4)` .

Options

A. `3400K`
B. `1012K`
C. `1000K`
D. `5700K`

Explanation

To solve the problem, we will use the Stefan-Boltzmann Law, which states that the radiant power (P) emitted by a black body is proportional to the fourth power of its absolute temperature (T). The formula is given by:

\[ P = \sigma A T^4 \]

Where:
- \( P \) = radiant power (in watts)
- \( \sigma \) = Stefan-Boltzmann constant (\( 5.7 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \))
- \( A \) = surface area (in square meters)
- \( T \) = absolute temperature (in Kelvin)

### Step-by-Step Solution:

**Step 1: Convert radiant power from kilowatts to watts.**
\[ P = 34.2 \, \text{kW} = 34.2 \times 10^3 \, \text{W} = 34200 \, \text{W} \]

**Step 2: Substitute the known values into the Stefan-Boltzmann equation.**
We have:
- \( P = 34200 \, \text{W} \)
- \( A = 0.6 \, \text{m}^2 \)
- \( \sigma = 5.7 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \)

Substituting these values into the equation:
\[ 34200 = (5.7 \times 10^{-8}) \times (0.6) \times T^4 \]

**Step 3: Simplify the equation to solve for \( T^4 \).**
First, calculate \( \sigma A \):
\[ \sigma A = (5.7 \times 10^{-8}) \times (0.6) = 3.42 \times 10^{-8} \]

Now, substituting this back into the equation:
\[ 34200 = 3.42 \times 10^{-8} \times T^4 \]

**Step 4: Rearrange the equation to isolate \( T^4 \).**
\[ T^4 = \frac{34200}{3.42 \times 10^{-8}} \]

**Step 5: Calculate \( T^4 \).**
\[ T^4 = \frac{34200}{3.42 \times 10^{-8}} = 1.000 \times 10^{12} \]

**Step 6: Take the fourth root to find \( T \).**
\[ T = (1.000 \times 10^{12})^{1/4} = 10^3 = 1000 \, \text{K} \]

### Final Answer:
The temperature of the furnace is \( 1000 \, \text{K} \).

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