A thin square steel plate with each side equal to 10 cm is heated by a blacksmith. The rate of radiated energy by the heated plate is 1134 W . The temperature of the hot steel plate is (Stefan's constant `sigma = 5.67 xx10^(-8) " watt "m^(-2) K^(-4)` , emissivity of the plate = 1)

Options

A. 1000 K
B. 1189 K
C. 2000 K
D. 2378 K

Answer

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

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

Where:
- \( P \) is the power radiated (in watts),
- \( A \) is the area of the plate (in square meters),
- \( \epsilon \) is the emissivity of the plate (dimensionless),
- \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2 \, \text{K}^4 \)),
- \( T \) is the absolute temperature (in Kelvin).

### Step-by-Step Solution:

1. **Calculate the Area of the Plate**:
The side of the square plate is given as 10 cm. First, we need to convert this to meters:
\[ \text{Side} = 10 \, \text{cm} = 0.1 \, \text{m} \]
Now, calculate the area \( A \):
\[ A = \text{Side}^2 = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

2. **Substitute Known Values into the Stefan-Boltzmann Equation**:
We know:
- \( P = 1134 \, \text{W} \)
- \( \epsilon = 1 \)
- \( \sigma = 5.67 \times 10^{-8} \, \text{W/m}^2 \, \text{K}^4 \)
Substitute these values into the equation:
\[ 1134 = 0.01 \cdot 1 \cdot (5.67 \times 10^{-8}) \cdot T^4 \]

3. **Simplify the Equation**:
Rearranging the equation gives:
\[ 1134 = 5.67 \times 10^{-10} \cdot T^4 \]

4. **Solve for \( T^4 \)**:
To isolate \( T^4 \), divide both sides by \( 5.67 \times 10^{-10} \):
\[ T^4 = \frac{1134}{5.67 \times 10^{-10}} \approx 2.000 \times 10^{12} \]

5. **Calculate \( T \)**:
Now, take the fourth root to find \( T \):
\[ T = (2.000 \times 10^{12})^{1/4} \]
Calculating this gives:
\[ T \approx 1189 \, \text{K} \]

### Final Answer:
The temperature of the hot steel plate is approximately **1189 K**.