Calculate the amount of radiant energy from a black body at a temperature of (i) `27^(@)`C (ii) `2727^(@)`C. `sigma = 5.67xx10^(-8)Wm^(-2)K^(-4)`.

Explanation

To calculate the amount of radiant energy from a black body at given temperatures, we will use the Stefan-Boltzmann Law, which states that the power radiated per unit area of a black body is directly proportional to the fourth power of its absolute temperature (in Kelvin). The formula is given by:

\[ E = \sigma T^4 \]

where:
- \( E \) is the radiant energy (in watts per square meter, W/m²),
- \( \sigma \) 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

#### Part (i): Temperature = 27°C

1. **Convert Celsius to Kelvin**:
\[ T = 27 + 273 = 300 \, \text{K} \]

2. **Apply the Stefan-Boltzmann Law**:
\[ E = \sigma T^4 = 5.67 \times 10^{-8} \times (300)^4 \]

3. **Calculate \( (300)^4 \)**:
\[ (300)^4 = 300^4 = 81 \times 10^8 \]

4. **Substitute back into the equation**:
\[ E = 5.67 \times 10^{-8} \times 81 \times 10^8 \]

5. **Simplify the expression**:
\[ E = 5.67 \times 81 \, \text{W/m}^2 \]
\[ E = 459.27 \, \text{W/m}^2 \]

#### Part (ii): Temperature = 2727°C

1. **Convert Celsius to Kelvin**:
\[ T = 2727 + 273 = 3000 \, \text{K} \]

2. **Apply the Stefan-Boltzmann Law**:
\[ E = \sigma T^4 = 5.67 \times 10^{-8} \times (3000)^4 \]

3. **Calculate \( (3000)^4 \)**:
\[ (3000)^4 = 81 \times 10^{12} \]

4. **Substitute back into the equation**:
\[ E = 5.67 \times 10^{-8} \times 81 \times 10^{12} \]

5. **Simplify the expression**:
\[ E = 5.67 \times 81 \times 10^{4} \, \text{W/m}^2 \]
\[ E = 459.27 \times 10^{4} \, \text{W/m}^2 \]
\[ E = 4.59 \times 10^{6} \, \text{W/m}^2 \]

### Final Answers:
- For 27°C: \( E = 459.27 \, \text{W/m}^2 \)
- For 2727°C: \( E = 4.59 \times 10^{6} \, \text{W/m}^2 \)