Two particles A and B , each carrying charge Q are held fixed with a separation D between them. A particle C having mass m and charge q is kept at the middle point of the line AB. (a) If it is displaced throught a distance x perpendicular to AB, what would be the electric force experienced by it .(b) Assuming x ltltd, show that this force is proportional to x. (c) under what conditions will the paricle C execute simple harmonic motion if it is released after such a small displacement? Find the time period of the oscillations if these conditions are satisfied.

Explanation

To solve the problem step by step, we will break it down into parts as per the question requirements.

### Step 1: Understanding the Setup
We have two fixed charges, A and B, each with charge \( Q \), separated by a distance \( D \). A third charge \( C \) with charge \( q \) and mass \( m \) is placed at the midpoint between A and B. When \( C \) is displaced a small distance \( x \) perpendicular to the line joining A and B, we need to find the electric force acting on it.

### Step 2: Calculate the Electric Forces
The electric force on charge \( C \) due to charge \( A \) (denoted as \( F_{AO} \)) and charge \( B \) (denoted as \( F_{BO} \)) can be calculated using Coulomb's law:

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