A particle of mass m and negative charge −q is constrained to move in a horizontal plane. In the situations described below, this particle can either oscillate back and forth in a straight line or move in a circle. (These two modes of motion are interesting because they generate linearly and circularly polarized radiation, respectively, but in this problem you may ignore any energy lost to radiation.)
a. A large positive charge Q≫q is fixed in place a distance R directly below the origin of the plane.
i. When the particle is a distance r≪R from the origin, find an approximate expression for its potential energy due to the charge Q to second order in r/R, up to an arbitrary constant. You may use this result for the rest of the problem.
ii. If the particle oscillates linearly with amplitude a≪R, what is its angular frequency ωℓ?
iii. If the particle performs circular motion with radius r≪R, what is its angular frequency ωc?
b. Now an additional negative charge −q is fixed in place at the origin of the plane.
i. What is the equilibrium distance r0 of the particle from the origin?
ii. If the particle oscillates linearly with amplitude a≪r0, what is its angular frequency Ωℓ? Is it higher or lower than ωℓ?
iii. Now suppose the particle performs circular motion with radius r=r0+δr, where δr≪r0. What is its angular frequency Ωc, in terms of ωc, r, and δr? Is it higher or lower than ωc?