A spherical tungsten pieces of radius `1.0cm` is suspended in an evacuated chamber maintained at `300K`. The pieces is maintained at 1000K by heating it electrically. Find the rate at which the electrical energy must be supplied. The emissivity of tungsten is `0.30` and the Stefan constant `sigma` is `6.0xx10^(-s)Wm^(-2)K^(-4)` .

Explanation

To solve the problem, we need to calculate the rate at which electrical energy must be supplied to maintain the temperature of the tungsten sphere at 1000 K. This can be done using the Stefan-Boltzmann law, which describes the power radiated by a black body in terms of its temperature.

### Step-by-step Solution:

1. **Identify Given Values:**
- Radius of the tungsten sphere, \( r = 1.0 \, \text{cm} = 0.01 \, \text{m} \)
- Temperature of the tungsten sphere, \( T_1 = 1000 \, \text{K} \)
- Temperature of the chamber, \( T_0 = 300 \, \text{K} \)
- Emissivity of tungsten, \( \epsilon = 0.30 \)
- Stefan-Boltzmann constant, \( \sigma = 6.0 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \)

2. **Calculate the Surface Area of the Sphere:**
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
Substituting the value of \( r \):
\[ A = 4 \pi (0.01)^2 = 4 \pi (0.0001) = 0.0004 \pi \, \text{m}^2 \approx 0.00125664 \, \text{m}^2 \]

3. **Calculate the Power Radiated by the Sphere:**
According to the Stefan-Boltzmann law, the power radiated by the sphere can be expressed as:
\[ P_{\text{radiated}} = \epsilon \sigma A (T_1^4 - T_0^4) \]
Substituting the known values:
\[ P_{\text{radiated}} = 0.30 \times (6.0 \times 10^{-8}) \times (0.0004 \pi) \times (1000^4 - 300^4) \]

4. **Calculate \( T_1^4 - T_0^4 \):**
First, calculate \( 1000^4 \) and \( 300^4 \):
\[ 1000^4 = 10^{12} \quad \text{and} \quad 300^4 = 8.1 \times 10^{9} \]
Therefore,
\[ T_1^4 - T_0^4 = 10^{12} - 8.1 \times 10^{9} \approx 10^{12} \quad (\text{since } 10^{12} \text{ is much larger}) \]

5. **Calculate the Power:**
Now substitute back into the power equation:
\[ P_{\text{radiated}} = 0.30 \times (6.0 \times 10^{-8}) \times (0.0004 \pi) \times (10^{12}) \]
\[ P_{\text{radiated}} = 0.30 \times (6.0 \times 10^{-8}) \times (0.0004 \times 3.14) \times (10^{12}) \]
\[ P_{\text{radiated}} \approx 0.30 \times (6.0 \times 10^{-8}) \times (0.0004 \times 3.14) \times (10^{12}) \approx 22.0 \, \text{W} \]

6. **Conclusion:**
The rate at which electrical energy must be supplied to maintain the temperature of the tungsten sphere at 1000 K is approximately **22 Watts**.