A ring of radius R that lies in the yz plane carries a positive charge Q uniformly distributed over its length. A particle of mass m that carries a negative charge of magnitude q is at the center of the ring.

(a) Show that if x << R, the electric field along the axis of the ring is proportional to x.

(b) Find the force on the mass m as a function of x.

(c) Show that if m is given a small displacement in the x direction, it will perform simple harmonic motion. Calculate the period of that motion.