Interaction between a point charge and a ring

A point charge q=+1 μCq=+1~\mu\mbox{C} with mass M=3 gM=3~\mbox{g} is placed at the center of a thin ring of radius R=10 cmR=10~\mbox{cm}, mass m=6 gm=6~\mbox{g}, and charge Q=5μCQ=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 v0\vec{v}_{0} in the plane of the ring. Determine the minimum time τ\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: 11+x2R22xRcos(θ)1+cos(θ)xR13cos2(θ)2x2R2\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 xR x \ll R .

Details and assumptions

  • k=14πϵ0=9×109 m/F.k=\frac{1}{4\pi \epsilon_{0}}= 9\times 10^{9}~\mbox{m/F}.

  • Neglect gravitational forces.

  • The ring is free to move.

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