The dimensional formula of the physical quantity whose S.I. unit is farad is

Options

A. `ML^(2)T^(3)I^(-2)`
B. `M^(-1)L^(-3)T^(4)I^(2)`
C. `M^(-1)L^(-2)T^(4)I^(2)`
D. `M^(-1)L^(-2)T^(3)I^(2)`

Answer

To find the dimensional formula of the physical quantity whose SI unit is farad (F), we need to understand what capacitance is and how it relates to charge and voltage.

### Step-by-Step Solution:

1. **Identify the Physical Quantity**:
The SI unit of capacitance is the farad (F). Capacitance (C) is defined as the ability of a system to store charge per unit voltage.

2. **Write the Formula for Capacitance**:
The formula for capacitance is given by:
\[ C = \frac{Q}{V} \]
where \( Q \) is the charge and \( V \) is the voltage.

3. **Express Voltage in Terms of Work**:
Voltage (or potential difference) can be expressed as:
\[ V = \frac{W}{Q} \]
where \( W \) is the work done.

4. **Substituting Voltage into the Capacitance Formula**:
Substituting the expression for voltage into the capacitance formula gives:
\[ C = \frac{Q}{\frac{W}{Q}} = \frac{Q^2}{W} \]

5. **Dimensional Formula for Charge (Q)**:
Charge is defined as current (I) multiplied by time (t):
\[ Q = I \cdot t \]
The dimensional formula for current (I) is \( A \) (amperes), so:
\[ [Q] = [I][t] = [A][T] = A \cdot T \]
Therefore, the dimensional formula for \( Q^2 \) is:
\[ [Q^2] = [A^2][T^2] \]

6. **Dimensional Formula for Work (W)**:
Work is defined as force (F) multiplied by distance (s):
\[ W = F \cdot s \]
The dimensional formula for force is:
\[ [F] = [M][L][T^{-2}] \]
Therefore, the dimensional formula for work is:
\[ [W] = [M][L][T^{-2}] \cdot [L] = [M][L^2][T^{-2}] \]

7. **Combine the Dimensional Formulas**:
Now substituting the dimensional formulas into the capacitance formula:
\[ [C] = \frac{[Q^2]}{[W]} = \frac{[A^2][T^2]}{[M][L^2][T^{-2}]} \]

8. **Simplifying the Expression**:
This simplifies to:
\[ [C] = \frac{A^2 T^2}{M L^2 T^{-2}} = A^2 T^4 M^{-1} L^{-2} \]

9. **Final Dimensional Formula**:
Thus, the final dimensional formula for capacitance (and hence for farad) is:
\[ [C] = M^{-1} L^{-2} T^4 A^2 \]

### Conclusion:
The dimensional formula of the physical quantity whose SI unit is farad is:
\[ [M^{-1} L^{-2} T^4 A^2] \]