Capacitance has dimensions :

Options

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

Answer

To find the dimensions of capacitance, we start with the relationship between charge (Q), capacitance (C), and potential difference (V):

1. **Understanding the relationship**:
The capacitance \( C \) is defined as the charge \( Q \) stored per unit potential \( V \):
\[ C = \frac{Q}{V} \]

2. **Identifying dimensions of charge**:
The charge \( Q \) can be expressed in terms of current \( I \) and time \( T \):
\[ Q = I \cdot T \]
In dimensional terms, charge has the dimensions:
\[ [Q] = [I][T] = [A][T] \]
where \( [A] \) is the dimension of current.

3. **Identifying dimensions of potential**:
The potential \( V \) can be expressed in terms of energy per unit charge. The energy \( E \) has the dimensions of:
\[ [E] = [M][L^2][T^{-2}] \]
Therefore, the potential \( V \) has the dimensions:
\[ [V] = \frac{[E]}{[Q]} = \frac{[M][L^2][T^{-2}]}{[A][T]} = [M][L^2][T^{-3}][A^{-1}] \]

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

5. **Simplifying the expression**:
Simplifying the above expression gives:
\[ [C] = \frac{[A^2][T^4]}{[M][L^2]} \]
Therefore, the dimensions of capacitance can be expressed as:
\[ [C] = [M^{-1}][L^{-2}][T^4][A^2] \]

6. **Final answer**:
Thus, the dimensions of capacitance are:
\[ [C] = [M^{-1}][L^{-2}][T^4][A^2] \]