A capacitor is given a charge `q`. The distance between the plates of the capacitor is `d`. One of the plates is fixed and the other plate is moved away from the other till the distance between them becomes `2d`. Find the work done by the external force.

Explanation

To find the work done by the external force when one plate of a capacitor is moved away from the other, we can follow these steps:

### Step 1: Understand the Initial Conditions
- We have a capacitor with charge \( q \) and plates separated by a distance \( d \).
- One plate is fixed, and the other plate is moved away until the distance becomes \( 2d \).

### Step 2: Determine the Force on the Plates
- The force between the plates of a capacitor can be derived from the electric field \( E \) between the plates and the charge \( q \) on the plates.
- The electric field \( E \) between the plates is given by:
\[ E = \frac{q}{\epsilon_0 A} \]
where \( \epsilon_0 \) is the permittivity of free space and \( A \) is the area of the plates.

- The force \( F \) on one plate due to the electric field from the other plate is:
\[ F = qE = q \left(\frac{q}{\epsilon_0 A}\right) = \frac{q^2}{\epsilon_0 A} \]

### Step 3: Calculate the Work Done
- The work done \( W \) by the external force when moving the plate from distance \( d \) to \( 2d \) is given by:
\[ W = F \cdot d_{\text{displacement}} \]
- The displacement \( d_{\text{displacement}} \) in this case is \( 2d - d = d \).

- Therefore, the work done by the external force is:
\[ W = F \cdot d = \left(\frac{q^2}{\epsilon_0 A}\right) \cdot d \]

### Step 4: Final Expression for Work Done
- Substituting the values, we get:
\[ W = \frac{q^2 d}{\epsilon_0 A} \]

This is the work done by the external force in moving the plate from distance \( d \) to \( 2d \).

### Summary
The work done by the external force when one plate of a capacitor is moved from distance \( d \) to \( 2d \) is:
\[ W = \frac{q^2 d}{\epsilon_0 A} \]