Positive charge Q is distributed uniformly over a circular ring of radius R. A particle having a mass m and a negative charge q, is placed on its axis at a distance x from the centre. Find the force on the particle. Assuming x is very less than R, find the time period of oscillation of the particle if it is released from there.

Explanation

To solve the problem step by step, we will first find the force acting on the particle due to the electric field created by the charged ring, and then we will determine the time period of oscillation for the particle when it is released from a small distance along the axis of the ring.

### Step 1: Determine the Electric Field due to the Ring

The electric field \( E \) at a distance \( x \) from the center of a uniformly charged ring of radius \( R \) and total charge \( Q \) is given by the formula:

\[
E = \frac{kQx}{(R^2 + x^2)^{3/2}}
...