A full radiator at `0^(@)C` radiates energy at the rate of `3.2 xx 10^(4) erg cm^(-2) s^(-1)`. Find (i) Stefan's constant and (ii) the amount of heat radiated per second by a sphere of radius 4 cm and at a temperature of `1000^(@)C`.

Answer

To solve the problem step by step, we will break it down into two parts as requested.

### Part (i): Finding Stefan's Constant

1. **Given Data**:
- Energy radiated per unit area at \(0^\circ C\) (which is \(273 K\)) is \(3.2 \times 10^4 \, \text{erg} \, \text{cm}^{-2} \, \text{s}^{-1}\).

2. **Conversion to Joules**:
- We know that \(1 \, \text{erg} = 10^{-7} \, \text{J}\).
- Therefore, convert the energy radiated to Joules:
\[ E = 3.2 \times 10^4 \, \text{erg} \, \text{cm}^{-2} \, \text{s}^{-1} \times 10^{-7} \, \text{J/erg} = 3.2 \times 10^{-3} \, \text{J} \, \text{cm}^{-2} \, \text{s}^{-1} \]

3. **Using Stefan-Boltzmann Law**:
- The Stefan-Boltzmann law states that the power radiated per unit area \(E\) is given by:
\[ E = \sigma T^4 \]
- Where \(\sigma\) is Stefan's constant and \(T\) is the absolute temperature in Kelvin.

4. **Substituting Values**:
- We have \(T = 273 \, K\) and \(E = 3.2 \times 10^{-3} \, \text{J} \, \text{cm}^{-2} \, \text{s}^{-1}\).
- Rearranging the equation to find \(\sigma\):
\[ \sigma = \frac{E}{T^4} \]
- Now substituting the values:
\[ \sigma = \frac{3.2 \times 10^{-3}}{(273)^4} \]

5. **Calculating \(T^4\)**:
- Calculate \(T^4\):
\[ T^4 = 273^4 = 5.578 \times 10^9 \, K^4 \]

6. **Final Calculation of \(\sigma\)**:
- Now substituting \(T^4\) into the equation:
\[ \sigma = \frac{3.2 \times 10^{-3}}{5.578 \times 10^9} \approx 5.76 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4} \]

### Part (ii): Amount of Heat Radiated by a Sphere

1. **Given Data**:
- Radius of the sphere \(r = 4 \, \text{cm} = 0.04 \, \text{m}\)
- Temperature \(T = 1000^\circ C = 1273 \, K\)

2. **Surface Area of the Sphere**:
- The surface area \(A\) of a sphere is given by:
\[ A = 4\pi r^2 \]
- Substituting the radius:
\[ A = 4\pi (0.04)^2 = 4\pi (0.0016) \approx 0.0201 \, \text{m}^2 \]

3. **Using Stefan-Boltzmann Law**:
- The total power \(P\) radiated by the sphere is given by:
\[ P = \sigma A T^4 \]

4. **Substituting Values**:
- We already calculated \(\sigma\) and now substitute \(A\) and \(T\):
\[ P = (5.76 \times 10^{-8}) \times (0.0201) \times (1273)^4 \]

5. **Calculating \(T^4\)**:
- Calculate \(T^4\):
\[ T^4 = 1273^4 \approx 2.573 \times 10^{12} \, K^4 \]

6. **Final Calculation of Power**:
- Now substituting all values:
\[ P \approx (5.76 \times 10^{-8}) \times (0.0201) \times (2.573 \times 10^{12}) \approx 3.041 \times 10^3 \, \text{W} \]

### Final Answers:
1. Stefan's constant \(\sigma \approx 5.76 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}\)
2. Amount of heat radiated per second by the sphere \(P \approx 3041 \, \text{W}\)