A ring of radius R carries a uniformly distributed charge +Q. A point charge `-q` is placed on the axis of the ring at a distance 2R from the centre of the ring and released from rest. The particle executes a simple harmonic motion along the axis of the ring. State True or False.

Explanation

Consider an element of the ring. Its enlarged view is as shown. For equilibrium of this segment, we can write. `F= 2 Delta T sin ((d theta)/(2))`
Here F is the repulsive force between q and elemental charge
`dQ [ because dQ = (Q)/(2pi R) (R d theta)]`
The electric outward force on element is `F= (1)/(4pi in_(0))(qdQ)/(R^(2))`
From the above three equations, we can write
`(1)/(4pi in_(0)) (q)/(R^(2)) (QR d theta)/(2pi R) ~~ 2 Delta T ((d theta)/(2)) ( because sin alpha ~~ alpha " for small angle")`
`Delta T= (Qq)/(8pi^(2) in_(0)R^(2))`