Dimensional formula of capacitance is

Options

A. `[A^(2)M^(-1)L^(-1)T^(4)]`
B. `[AM^(-1)L^(-2)T^(4)]`
C. `[A^(2)M^(-2)L^(-2)T^(4)]`
D. `[A^(0)M^(0)L^(-2)T^(4)]`

Answer

To find the dimensional formula of capacitance, we start with the relationship between charge (Q), capacitance (C), and potential difference (V). The formula is given by:

\[ Q = C \cdot V \]

From this, we can rearrange the equation to express capacitance (C) in terms of charge and potential difference:

\[ C = \frac{Q}{V} \]

Next, we need to find the dimensional formulas for charge (Q) and potential difference (V).

1. **Finding the Dimensional Formula of Charge (Q)**:
- Charge is defined as the product of current (I) and time (t).
- The dimensional formula for current (I) is \([A]\) (Ampere).
- Therefore, the dimensional formula for charge (Q) is:
\[ Q = I \cdot t \implies [Q] = [A] \cdot [T] = A \cdot T \]

2. **Finding the Dimensional Formula of Potential Difference (V)**:
- Potential difference (V) is defined as work done (W) per unit charge (Q).
- Work (W) is defined as force (F) multiplied by displacement (s). The dimensional formula for force is:
\[ F = m \cdot a = [M] \cdot \left[\frac{L}{T^2}\right] = M \cdot L \cdot T^{-2} \]
- Therefore, the dimensional formula for work (W) is:
\[ W = F \cdot s \implies [W] = [M] \cdot [L] \cdot [T^{-2}] \cdot [L] = M \cdot L^2 \cdot T^{-2} \]
- Now, substituting the dimensional formula for work and charge into the formula for potential difference:
\[ V = \frac{W}{Q} \implies [V] = \frac{[W]}{[Q]} = \frac{M \cdot L^2 \cdot T^{-2}}{A \cdot T} = M \cdot L^2 \cdot T^{-3} \cdot A^{-1} \]

3. **Substituting into the Capacitance Formula**:
- Now we substitute the dimensional formulas of Q and V back into the capacitance formula:
\[ C = \frac{Q}{V} \implies [C] = \frac{[Q]}{[V]} = \frac{A \cdot T}{M \cdot L^2 \cdot T^{-3} \cdot A^{-1}} = \frac{A^2 \cdot T^4}{M \cdot L^2} \]

Thus, the dimensional formula of capacitance (C) is:

\[ [C] = \frac{A^2 \cdot T^4}{M \cdot L^2} \]

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