What is the S.I. unit of Stefan-Boltzmann's constant ?

Options

A. `Wm^(-2)K^(-4)`
B. `Wm^(2)K^(4)`
C. `WK^(-4)`
D. erg `s^(-2)K^(-4)`

Answer

To find the S.I. unit of Stefan-Boltzmann's constant (σ), we can start from the Stefan-Boltzmann law, which states that the total energy radiated per unit surface area of a black body per unit time (Q) is proportional to the fourth power of the black body's absolute temperature (T). The formula can be expressed as:

\[ Q = \sigma T^4 \]

Where:
- \( Q \) is the energy emitted per unit time per unit area (in watts per square meter, \( \text{W/m}^2 \)).
- \( \sigma \) is the Stefan-Boltzmann constant.
- \( T \) is the absolute temperature (in Kelvin, K).

### Step-by-Step Solution:

1. **Identify the units of Q**:
- The energy emitted per unit time per unit area is measured in watts per square meter.
- Therefore, the unit of \( Q \) is \( \text{W/m}^2 \).

2. **Identify the units of T**:
- The temperature \( T \) is measured in Kelvin (K).

3. **Express the formula in terms of units**:
- From the formula \( Q = \sigma T^4 \), we can rearrange it to find the units of \( \sigma \):
\[ \sigma = \frac{Q}{T^4} \]

4. **Substitute the units into the equation**:
- Substitute the unit of \( Q \) and the unit of \( T \):
\[ \sigma = \frac{\text{W/m}^2}{K^4} \]

5. **Simplify the units**:
- This can be rewritten as:
\[ \sigma = \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4} \]

6. **Final result**:
- Therefore, the S.I. unit of Stefan-Boltzmann's constant is:
\[ \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4} \]

### Conclusion:
The S.I. unit of Stefan-Boltzmann's constant (σ) is \( \text{W} \cdot \text{m}^{-2} \cdot \text{K}^{-4} \).