Which of the following represents the dimensions of Farad

Options

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

Answer

To find the dimensions of the unit Farad, which is the unit of capacitance, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Definition of Capacitance**:
The capacitance \( C \) is defined by the formula:
\[ C = \frac{Q}{V} \]
where \( Q \) is the charge and \( V \) is the voltage.

2. **Express Voltage in Terms of Work**:
Voltage \( V \) can be defined as work done per unit charge:
\[ V = \frac{W}{Q} \]
Therefore, we can rewrite the capacitance formula as:
\[ C = \frac{Q}{\frac{W}{Q}} = \frac{Q^2}{W} \]

3. **Substitute the Dimensions of Charge and Work**:
- The dimension of charge \( Q \) can be expressed in terms of current \( I \) and time \( T \):
\[ Q = I \cdot T \quad \text{(where } I \text{ is in Amperes)} \]
- The dimension of work \( W \) is given by force times displacement. Force can be expressed as mass times acceleration:
\[ W = \text{Force} \times \text{Displacement} = (M \cdot L \cdot T^{-2}) \cdot L = M \cdot L^2 \cdot T^{-2} \]

4. **Substituting Dimensions into the Capacitance Formula**:
Now substituting the dimensions into the capacitance formula:
\[ C = \frac{(I \cdot T)^2}{M \cdot L^2 \cdot T^{-2}} \]
This simplifies to:
\[ C = \frac{I^2 \cdot T^2}{M \cdot L^2 \cdot T^{-2}} = \frac{I^2 \cdot T^4}{M \cdot L^2} \]

5. **Final Dimensions of Farad**:
Therefore, the dimensions of Farad can be expressed as:
\[ [C] = M^{-1} \cdot L^{-2} \cdot T^4 \cdot I^2 \]

### Final Answer:
The dimensions of Farad are:
\[ \text{Farad} = M^{-1} \cdot L^{-2} \cdot T^4 \cdot I^2 \]