A blackened solid copper sphere of radius 2 cm is placed in an evacuated enclosure whose walls are kept at `100^(@)C`. At what rate must energy be supplied to the sphere to keep its temperature constant at `127^(@)C` ? Stefan constant `= 5.67xx 10^(-8)Jm^(-2)K^(-4).`

Answer

To solve the problem, we need to determine the rate at which energy must be supplied to a blackened solid copper sphere to maintain its temperature at 127°C when placed in an evacuated enclosure with walls at 100°C. We will use the Stefan-Boltzmann law for this calculation.

### Step-by-Step Solution:

1. **Convert Temperatures to Kelvin**:
- The temperature of the sphere, \( T = 127°C = 127 + 273 = 400 K \).
- The temperature of the walls, \( T_0 = 100°C = 100 + 273 = 373 K \).

2. **Calculate the Surface Area of the Sphere**:
- The radius of the sphere, \( r = 2 cm = 0.02 m \).
- The surface area \( A \) of a sphere is given by the formula:
\[ A = 4 \pi r^2 \]
- Substituting the radius:
\[ A = 4 \pi (0.02)^2 = 4 \pi (0.0004) = 0.005024 \, m^2 \]

3. **Apply the Stefan-Boltzmann Law**:
- The rate of energy radiated (or absorbed) is given by:
\[ E = \sigma A (T^4 - T_0^4) \]
- Where \( \sigma = 5.67 \times 10^{-8} \, J m^{-2} K^{-4} \).

4. **Calculate \( T^4 \) and \( T_0^4 \)**:
- Calculate \( T^4 \):
\[ T^4 = (400)^4 = 256 \times 10^{8} = 2.56 \times 10^{10} \, K^4 \]
- Calculate \( T_0^4 \):
\[ T_0^4 = (373)^4 = 1.93 \times 10^{10} \, K^4 \]

5. **Substitute Values into the Formula**:
- Now substitute the values into the energy formula:
\[ E = 5.67 \times 10^{-8} \times 0.005024 \times (2.56 \times 10^{10} - 1.93 \times 10^{10}) \]
- Calculate the difference:
\[ 2.56 \times 10^{10} - 1.93 \times 10^{10} = 0.63 \times 10^{10} \]

6. **Final Calculation**:
- Now substituting back:
\[ E = 5.67 \times 10^{-8} \times 0.005024 \times 0.63 \times 10^{10} \]
- Calculate:
\[ E = 5.67 \times 0.005024 \times 0.63 \times 10^{2} \, J/s \]
\[ E \approx 1.79 \, J/s \]

### Final Answer:
The rate at which energy must be supplied to the sphere to keep its temperature constant at 127°C is approximately **1.79 J/s**.