Two identical particles, each having charge +*q*, are fixed in space and separated by a distance *d*. A third particle with charge −*Q* is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance *x* from the midpoint between those charges (see figure below).

(a) Show that if *x* is small compared with *d*, the motion of −*Q* is simple harmonic along the perpendicular bisector. (Do this on paper. Your instructor may ask you to turn in this work.)

(b) Determine the period of that motion. (Use the following as necessary: *π*, *q*, *Q*, *m* for the mass of charge *Q*, *d*, and *k _{e}*.)

*T*=

(c) How fast will the charge −

*Q*be moving when it is at the midpoint between the two fixed charges if initially it is released at a distance

*a*<<

*d*from the midpoint? (Use the following as necessary:

*π*,

*q*,

*Q*,

*m*for the mass of charge

*Q*,

*d*, and

*k*.)

_{e}*v*=