The dimensional formula for stefan's constant 's' is

Options

A. (a)`M^(1)L^(0)T^(-3)K^(-4)`
B. (b)`M^(1)L^(2)T^(-3)K^(-4)`
C. (c)`M^(0)L^(0)T^(-3)K^(-4)`
D. (d)`M^(1)L^(2)T^(-2)K^(-1)`

Answer

To find the dimensional formula for Stefan's constant (σ), we start with its definition in terms of physical quantities. The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body per unit time (also known as the Stefan's constant) is proportional to the fourth power of the black body's absolute temperature.

1. **Understanding the formula**:
The formula for Stefan's constant can be expressed as:
\[ \sigma = \frac{E}{A \cdot t \cdot T^4} \]
where:
- \(E\) is energy,
- \(A\) is area,
- \(t\) is time,
- \(T\) is temperature.

2. **Dimensional formula for energy (E)**:
The dimensional formula for energy is given by:
\[ [E] = [M][L^2][T^{-2}] \]
where:
- \(M\) is mass,
- \(L\) is length,
- \(T\) is time.

3. **Dimensional formula for area (A)**:
The area is given by:
\[ [A] = [L^2] \]

4. **Dimensional formula for time (t)**:
The dimensional formula for time is:
\[ [t] = [T] \]

5. **Dimensional formula for temperature (T)**:
The dimensional formula for temperature is denoted as:
\[ [T] = [K] \]

6. **Substituting the dimensional formulas into the equation**:
Now substituting these into the formula for Stefan's constant:
\[ \sigma = \frac{[M][L^2][T^{-2}]}{[L^2] \cdot [T] \cdot [K^4]} \]

7. **Simplifying the expression**:
When we simplify this, we have:
\[ \sigma = \frac{[M][L^2][T^{-2}]}{[L^2][T][K^4]} = \frac{[M][L^2][T^{-2}]}{[L^2][T^1][K^4]} \]
The \(L^2\) terms cancel out:
\[ \sigma = [M][T^{-2}][T^{-1}][K^{-4}] = [M][T^{-3}][K^{-4}] \]

8. **Final dimensional formula**:
Thus, the dimensional formula for Stefan's constant is:
\[ [\sigma] = [M^1][T^{-3}][K^{-4}] \]

9. **Conclusion**:
Therefore, the correct option for the dimensional formula of Stefan's constant is:
\[ [M^1][T^{-3}][K^{-4}] \]