A circular ring of radius R with uniform positive charge density `lambda` per unit length is located in the y z plane with its center at the origin O. A particle of mass m and positive charge q is projected from that point `p( - sqrt(3) R, 0,0)` on the negative x - axis directly toward O, with initial speed V. Find the smallest (nonzero) value of the speed such that the particle does not return to P ?

Explanation

To solve the problem, we need to find the minimum speed \( V \) of a charged particle projected towards a charged ring such that it does not return to its initial position \( P(-\sqrt{3}R, 0, 0) \).

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a circular ring of radius \( R \) with a uniform positive charge density \( \lambda \).
- The ring is located in the \( yz \) plane, centered at the origin \( O \).
- A particle of mass \( m \) and charge \( q \) is projected from point \( P(-\sqrt{3}R, 0, 0) \) towards the origin with an initial speed \( V \).
...