Arc in a dipole field

Under the influence of an electrical field that's caused by a dipole made of two opposite charges \(q_+=1~\mbox{C}\) and \(q_-=-1~\mbox{C}\) that are fixed in space and separated by a distance \(l=1~\mbox{mm}\), a test particle of mass \(m=1~\mbox{kg}\) and charge \(Q=5~\mbox{C}\) can oscillate back and forth with a trajectory that looks like half of a circle of radius \(R=1~\mbox{m}\). Find the time period of that oscillation in ms, given that the Coulomb constant is \(k=8.99 \times 10^9~\mbox{Nm}^2/\mbox{C}^2\).

Details and assumptions

• Hint: \(\int^{\pi/2}_{0} \frac{dx}{\sqrt{\cos{x}}}= 2.622\)