A particle of mass m and charge q is located midway between two fixed charged particles each having a charge q and a distance 2l apart. Prove that the motion of the particle will be SHM if it is displaced slightly along the line connecting them and released. Also find its time period.

Explanation

To solve the problem, we need to analyze the forces acting on the particle of mass \( m \) and charge \( q \) located midway between two fixed charged particles, each with charge \( q \) and separated by a distance \( 2l \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the two fixed charges be \( Q_1 \) and \( Q_2 \), each with charge \( q \).
- The distance between \( Q_1 \) and \( Q_2 \) is \( 2l \).
- The particle of mass \( m \) and charge \( q \) is located at the midpoint, which is at a distance \( l \) from each charge.
...