Question

A circular ring of radius R with uniform positive charge density λ per unit length is located in the y-z plane with its centre at the origin O, A particle of mass m and positive charge q is projected from the point P(R√3 , 0,0) on the positive x-axis directly towards O, with an initial speed v. Find the smallest (nonzero) value of the speed v such that the particle does not return to P.

Answer 1

Let the particle be at a distance x from the origin at a point P at any instant. The electric field at P due to charge on the circular ring is 

Small amount of work done in moving the particle toward O for an infinitsinally small distance dx dW = Fdx

This work done needs to be supplied by the kinetic energy given to the charge initially.

This is the minimum velocity through which the charge q must be projected to reach the origin.