Let *Q* be the charge on the ring, the negative charge -*q* is released from point ( 0, 0, *z*_{0} ). The electric field at *P* due to the charged ring will be along positive *z*-axis and its magnitude will be

.

E=Qz0/14πε0(R2+z20)^{3/2}

*E* = 0 at centre of the ring because *z*_{0} = 0

Force on charge at *P* will be towards centre as shown, and its magnitude is

F0=qE= Qqz0/14πε0(R2+z20)^{3/2} ..........(i)

Similarly, when it crosses the origin, the force is again towards centre *O*.

Thus, the motion of the particle is peroidic for all values of *z*_{0} lying between 0 and ∞.

Secondly, if z0<<R,(R^{2}+z_{0}^{2})3/2≈R^{3}

F0≈Qqz_{0}/14πε0⋅R^{3} [From Equation. ( i)]

i.e., the restoring force Fe∝−z0. Hence, the motion of the particle will be simple harmonic. (Here negative sign implies that the force is towards its mean position.)