# Interaction between a point charge and a ring

**Electricity and Magnetism**Level 5

A point charge \(q=+1~\mu\mbox{C}\) with mass \(M=3~\mbox{g}\) is placed at the center of a thin ring of radius \(R=10~\mbox{cm}\), mass \(m=6~\mbox{g}\), and charge \(Q=5 \mu\mbox{C} \). Initially the system is at rest in equilibrium. Then, the point charge is given a push so that it acquires a small velocity \(\vec{v}_{0}\) in the plane of the ring. Determine the minimum time \(\tau \) **in seconds** after which the point charge will be at the center of the ring again. Assume that the charge in the ring is distributed uniformly.
The following expansion might be useful:
\[\frac{1}{\sqrt{1+\frac{x^{2}}{R^{2}}-2 \frac{x}{R}\cos(\theta)}}\approx 1+\cos(\theta) \frac{x}{R}-\frac{1-3 \cos^{2}(\theta)}{2}\frac{x^{2}}{R^{2}}\]
for \( x \ll R \).

**Details and assumptions**

\[k=\frac{1}{4\pi \epsilon_{0}}= 9\times 10^{9}~\mbox{m/F}\]

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