A black body radiates 3 joule per square centimeter per second when its temperature is `127^(@)C`.How much heat will be radiated per square centimetre per second when its temperature is `527^(@)C`?

Options

A. 6 J
B. 12 J
C. 24 J
D. 48 J

Explanation

To solve the problem, we will use the Stefan-Boltzmann Law, which states that the power radiated by a black body per unit area is proportional to the fourth power of its absolute temperature. The formula is given by:

\[ P = \sigma T^4 \]

Where:
- \( P \) is the power radiated per unit area (in joules per square centimeter per second),
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin.

### Step-by-Step Solution:

1. **Convert the temperatures from Celsius to Kelvin**:
- For the first temperature \( T_1 = 127^\circ C \):
\[ T_1 = 127 + 273 = 400 \, K \]
- For the second temperature \( T_2 = 527^\circ C \):
\[ T_2 = 527 + 273 = 800 \, K \]

2. **Use the Stefan-Boltzmann Law for the first temperature**:
- Given that the power radiated at \( T_1 \) is \( P_1 = 3 \, \text{J/cm}^2/s \):
\[ P_1 = \sigma T_1^4 \]
- Substitute \( T_1 \):
\[ 3 = \sigma (400)^4 \]

3. **Calculate \( T_1^4 \)**:
- Calculate \( 400^4 \):
\[ 400^4 = (400 \times 400) \times (400 \times 400) = 160000 \times 160000 = 25600000000 \]

4. **Express \( \sigma \)**:
- Rearranging the equation gives:
\[ \sigma = \frac{3}{(400)^4} = \frac{3}{25600000000} \]

5. **Use the Stefan-Boltzmann Law for the second temperature**:
- For \( T_2 \):
\[ P_2 = \sigma T_2^4 \]
- Substitute \( T_2 \):
\[ P_2 = \sigma (800)^4 \]

6. **Calculate \( T_2^4 \)**:
- Calculate \( 800^4 \):
\[ 800^4 = (800 \times 800) \times (800 \times 800) = 640000 \times 640000 = 409600000000 \]

7. **Substituting \( \sigma \) into \( P_2 \)**:
- Substitute \( \sigma \):
\[ P_2 = \frac{3}{25600000000} \times 409600000000 \]

8. **Calculate \( P_2 \)**:
- Simplifying gives:
\[ P_2 = 3 \times \frac{409600000000}{25600000000} = 3 \times 16 = 48 \, \text{J/cm}^2/s \]

### Final Answer:
The heat radiated per square centimeter per second when the temperature is \( 527^\circ C \) is **48 Joules**.