Dimensional formula of capacitance is

Options

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

Explanation

To find the dimensional formula of capacitance, we start with the definition of capacitance and derive the necessary dimensions step by step.

### Step-by-Step Solution:

1. **Definition of Capacitance**:
Capacitance (C) is defined as the charge (Q) per unit potential difference (V):
\[ C = \frac{Q}{V} \]

2. **Dimension of Charge (Q)**:
Charge can be expressed in terms of current (I) and time (T):
\[ Q = I \cdot T \]
The dimension of current (I) is denoted as [A] (Ampere), and the dimension of time (T) is [T]. Therefore, the dimension of charge (Q) is:
\[ [Q] = [I][T] = [A][T] \]

3. **Dimension of Potential Difference (Voltage, V)**:
Voltage is defined as energy (W) per unit charge (Q):
\[ V = \frac{W}{Q} \]
The dimension of energy (W) is given by:
\[ [W] = [M][L^2][T^{-2}] \]
where [M] is mass, [L] is length, and [T] is time. Now substituting the dimension of charge (Q):
\[ [V] = \frac{[W]}{[Q]} = \frac{[M][L^2][T^{-2}]}{[A][T]} = \frac{[M][L^2][T^{-2}]}{[A][T]} = [M][L^2][T^{-3}][A^{-1}] \]

4. **Substituting Back into the Capacitance Formula**:
Now substituting the dimensions of charge (Q) and voltage (V) back into the capacitance formula:
\[ C = \frac{Q}{V} = \frac{[A][T]}{[M][L^2][T^{-3}][A^{-1}]} \]

5. **Simplifying the Expression**:
Now we simplify the expression:
\[ C = \frac{[A][T]}{[M][L^2][T^{-3}][A^{-1}]} = \frac{[A]^2[T^4]}{[M][L^2]} \]
Thus, the dimensional formula of capacitance (C) is:
\[ [C] = [M^{-1}][L^{-2}][T^4][A^2] \]

### Final Result:
The dimensional formula of capacitance is:
\[ [C] = [M^{-1} L^{-2} T^4 A^2] \]