A black body radiates 20 W at temperature `227^(@)C`. If temperature of the black body is changed to `727^(@)C` then its radiating power wil be

Options

A. 120 W
B. 240 W
C. 320 W
D. 360 W

Answer

To solve the problem, we will 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.

### Step-by-Step Solution:

1. **Convert temperatures from Celsius to Kelvin**:
- The formula to convert Celsius to Kelvin is:
\[ T(K) = T(°C) + 273 \]
- For the initial temperature \( T_1 = 227°C \):
\[ T_1 = 227 + 273 = 500 \, K \]
- For the final temperature \( T_2 = 727°C \):
\[ T_2 = 727 + 273 = 1000 \, K \]

2. **Use the Stefan-Boltzmann law**:
- The power radiated by a black body is given by:
\[ P \propto T^4 \]
- This can be expressed as:
\[ \frac{P_1}{P_2} = \frac{T_1^4}{T_2^4} \]
- Where \( P_1 = 20 \, W \) (the initial power) and \( P_2 \) is the power we want to find.

3. **Substituting the known values**:
- Plugging in the values we have:
\[ \frac{20}{P_2} = \frac{(500)^4}{(1000)^4} \]

4. **Simplifying the equation**:
- Simplifying the right side:
\[ \frac{(500)^4}{(1000)^4} = \frac{500^4}{1000^4} = \left(\frac{500}{1000}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]
- Thus, we have:
\[ \frac{20}{P_2} = \frac{1}{16} \]

5. **Cross-multiplying to find \( P_2 \)**:
- Cross-multiplying gives:
\[ 20 \cdot 16 = P_2 \]
- Therefore:
\[ P_2 = 320 \, W \]

### Final Answer:
The radiating power when the temperature of the black body is changed to \( 727°C \) will be **320 W**.