What are the dimensions of Stefan's constant ?

Options

A. `ML^(-2)T^(-2)K^(-4)`
B. `ML^(-1)T^(-2)K^(-4)`
C. `MLT^(-3)K^(-4)`
D. `ML^(0)T^(-3)K^(-4)`

Explanation

To find the dimensions of Stefan's constant (σ), we start from Stefan-Boltzmann law, which states:

\[ P = \sigma A E T^4 \]

Where:
- \( P \) is the power emitted (in watts),
- \( \sigma \) is Stefan's constant,
- \( A \) is the area of the body,
- \( E \) is the emissivity (dimensionless),
- \( T \) is the absolute temperature (in Kelvin).

### Step 1: Identify the dimensions of each term in the equation

1. **Power (P)**: The dimension of power is given by:
\[ [P] = [\text{Energy}] / [\text{Time}] = [ML^2T^{-2}] / [T] = [ML^2T^{-3}] \]

2. **Area (A)**: The dimension of area is:
\[ [A] = [L^2] \]

3. **Emissivity (E)**: Emissivity is dimensionless, so:
\[ [E] = 1 \]

4. **Temperature (T)**: The dimension of temperature is:
\[ [T] = [K] \]

### Step 2: Substitute the dimensions into the equation

Now we can rewrite the equation in terms of dimensions:
\[ [P] = [\sigma] [A] [E] [T^4] \]

Substituting the dimensions we found:
\[ [ML^2T^{-3}] = [\sigma] [L^2] [1] [K^4] \]

### Step 3: Rearranging for σ

To isolate σ, we rearrange the equation:
\[ [\sigma] = \frac{[ML^2T^{-3}]}{[L^2][K^4]} \]

### Step 4: Simplifying the dimensions

Now we simplify the right-hand side:
\[ [\sigma] = \frac{[ML^2T^{-3}]}{[L^2]} \cdot \frac{1}{[K^4]} = [M][L^{2-2}][T^{-3}][K^{-4}] \]
\[ [\sigma] = [M][T^{-3}][K^{-4}] \]

### Conclusion

Thus, the dimensions of Stefan's constant (σ) are:
\[ [\sigma] = [M^1 L^0 T^{-3} K^{-4}] \]