You are working for a manufacturing company. Your supervisor has an idea for controlling
the position of a small bead by using electric fields. The physical setup is shown in the
figure below.

Three fixed beads are secured along the y-axis, with a bead of charge +nq at
the origin and charges of −mq a distance a above and below the origin. The values of m and
n can be set by an operator. On an insulating wire laid along the x-axis, a movable bead with
charge −pq is placed. By adjusting m and n, the movable bead can be moved to different
equilibrium positions x along the wire. Your supervisor asks you to set up the system and
test it. In particular, he asks for answers to the following questions.

a) (5 marks) If n = 3 in a particular test, what is the value of m that will place the
movable bead in equilibrium at x = a?

b) (5 marks If n = 3, and the largest you can make m is m=7, what is the closest
equilibrium position of the bead to the origin? (Give your answer in terms of a)?