Dimensional Formula of Capacitance

 

❓ Question

Which one of the following is the correct dimensional formula for capacitance in farad (F)?

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✍️ Solution

Capacitance is defined as

$$C=\frac{Q}{V}$$

where

$$Q=\text{charge}, \qquad V=\text{potential difference}$$

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Step 1: Dimension of potential difference

We know

$$V=\frac{W}{Q}$$

Therefore,

$$[V]=\frac{[W]}{[Q]}$$

Now,

$$[W]=[ML^2T^{-2}]$$

Hence,

$$[V]=\frac{[ML^2T^{-2}]}{[Q]}$$

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Step 2: Dimension of capacitance

Using

$$C=\frac{Q}{V}$$
$$[C] = \frac{[Q]} {\dfrac{[ML^2T^{-2}]}{[Q]}}$$
$$= \frac{[Q]^2}{[ML^2T^{-2}]}$$

If $C$ denotes the dimension of charge, then

$$[F] = [C^2M^{-1}L^{-2}T^2]$$

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✅ Final Answer

$$\boxed{[F]=[C^2M^{-1}L^{-2}T^2]}$$

Therefore, the correct choice is

$$\boxed{\text{Option (1)}}$$

Key Idea

Since

$$1\text{ farad}=\frac{\text{coulomb}}{\text{volt}},$$

the dimensional formula in terms of charge $C$ is

$$\boxed{C^2M^{-1}L^{-2}T^2}$$