A spherical black body of 5 cm radius is maintained at a temperature of `327^(@)C`. Then the power radiated will be `(sigma=5.7xx10^(-8)"SI unit")`

Options

A. 58 W
B. 231 W
C. 75 W
D. 482 W

Explanation

To solve the problem of finding the power radiated by a spherical black body of radius 5 cm at a temperature of 327°C, we can follow these steps:

### Step-by-step Solution:

1. **Convert the Temperature to Kelvin**:
The temperature is given in degrees Celsius, so we need to convert it to Kelvin using the formula:
\[ T(K) = T(°C) + 273 \]
For our case:
\[ T = 327 + 273 = 600 \, K \]

2. **Identify the Area of the Sphere**:
The surface area \(A\) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
Here, the radius \(r\) is given as 5 cm, which we need to convert to meters:
\[ r = 5 \, \text{cm} = 0.05 \, \text{m} \]
Now, substituting the radius into the area formula:
\[ A = 4\pi (0.05)^2 = 4\pi (0.0025) = 0.0314 \, \text{m}^2 \, (\text{approximately}) \]

3. **Use the Stefan-Boltzmann Law**:
The power radiated by a black body is given by the Stefan-Boltzmann law:
\[ P = \sigma A T^4 \]
where:
- \(P\) is the power radiated,
- \(\sigma\) is the Stefan-Boltzmann constant, \(5.7 \times 10^{-8} \, \text{W/m}^2\text{K}^4\),
- \(A\) is the surface area,
- \(T\) is the temperature in Kelvin.

4. **Substitute the Values**:
Now we can substitute the values into the formula:
\[ P = (5.7 \times 10^{-8}) \times (0.0314) \times (600)^4 \]

5. **Calculate \(600^4\)**:
First, calculate \(600^4\):
\[ 600^4 = 129600000000 \]

6. **Calculate the Power**:
Now substituting back into the power equation:
\[ P = (5.7 \times 10^{-8}) \times (0.0314) \times (129600000000) \]
\[ P \approx (5.7 \times 10^{-8}) \times (0.0314) \times (1.296 \times 10^{11}) \]
\[ P \approx 2.31 \, \text{W} \]

### Final Answer:
The power radiated by the spherical black body is approximately \(2.31 \, \text{W}\).