You have a summer intern position with a company that designs and builds nanomachines. An engineer with the company is designing a microscopic oscillator to help keep time, and you€™ve been assigned to help him analyze the design. He wants to place a negative charge at the center of a very small, positively charged metal ring. His claim is that the negative charge will undergo simple harmonic motion at a frequency determined by the amount of charge on the ring.
a. Consider a negative charge near the center of a positively charged ring centered on the z-axis. Show that there is a restoring force on the charge if it moves along the z-axis but stays close to the center of the ring. That is, show there€™s a force that tries to keep the charge at z = 0.
b. Show that for small oscillations, with amplitude << R, a particle of mass m with charge -q undergoes simple harmonic motion with frequency

R and Q are the radius and charge of the ring.
c. Evaluate the oscillation frequency for an electron at the center of a 2.0 μm diameter ring charged to 1.0 × 10^-13 C.

qQ 4R f= 2