The emissivity of tungsten is 0.4. A tungsten sphere with a radius of 4.0 cm is suspended within a large evacuated enclosure whose walls are at 300 K. What power input is required to maintain the sphere at a temperature of 3000 K if heat conduction along supports is neglected ?
Take, `sigma = 5.67 xx (10^-8) Wm^2-K^4`.

Explanation

To solve the problem, we need to calculate the power input required to maintain a tungsten sphere at a temperature of 3000 K while it is placed in an evacuated enclosure with walls at 300 K. The emissivity of tungsten is given as 0.4, and we will use the Stefan-Boltzmann law for radiation.

### Step-by-Step Solution:

1. **Identify the given values:**
- Emissivity of tungsten, \( e = 0.4 \)
- Radius of the tungsten sphere, \( r = 4.0 \, \text{cm} = 0.04 \, \text{m} \)
- Temperature of the sphere, \( T = 3000 \, \text{K} \)
- Temperature of the enclosure, \( T_0 = 300 \, \text{K} \)
- Stefan-Boltzmann constant, \( \sigma = 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \)

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 radius:
\[ A = 4\pi (0.04)^2 = 4\pi (0.0016) \approx 0.0201 \, \text{m}^2 \]

3. **Apply the Stefan-Boltzmann law:**
The power emitted by the sphere can be calculated using the formula:
\[ P = e \sigma A (T^4 - T_0^4) \]

4. **Calculate \( T^4 \) and \( T_0^4 \):**
- \( T^4 = (3000)^4 = 8.1 \times 10^{12} \, \text{K}^4 \)
- \( T_0^4 = (300)^4 = 8.1 \times 10^{6} \, \text{K}^4 \)

5. **Calculate the difference \( T^4 - T_0^4 \):**
\[ T^4 - T_0^4 = 8.1 \times 10^{12} - 8.1 \times 10^{6} \approx 8.1 \times 10^{12} \, \text{K}^4 \]

6. **Substitute the values into the power equation:**
\[ P = 0.4 \times (5.67 \times 10^{-8}) \times (0.0201) \times (8.1 \times 10^{12}) \]

7. **Calculate the power \( P \):**
\[ P \approx 0.4 \times 5.67 \times 10^{-8} \times 0.0201 \times 8.1 \times 10^{12} \]
\[ P \approx 0.4 \times 5.67 \times 0.0201 \times 8.1 \times 10^{4} \, \text{W} \]
\[ P \approx 0.4 \times 9.157 \times 10^{4} \approx 3.663 \times 10^{4} \, \text{W} \]

8. **Final Result:**
The power input required to maintain the sphere at a temperature of 3000 K is approximately:
\[ P \approx 3.66 \times 10^{4} \, \text{W} \]