Hole in a plane **

(a) A hole of radius R is cut out from a very large flat sheet with uniform charge density σ . Let L be the line perpendicular to the sheet, passing through the center of the hole. What is the electric field at a point on L, a distance z from the center of the hole? Hint: Consider the plane to consist of many concentric rings.

(b) If a charge -q with mass m is released from rest on L, very close to the center of the hole, show that it undergoes oscillatory motion, and find the frequency ω of these oscillations. What is ω if m = 1 g, -q = -10-8 C, σ = 10-6 C/m2, and R = 0.1 m?

(c) If a charge -q with mass m is released from rest on L, a dis- tance z from the sheet, what is its speed when it passes through
the center of the hole? What does your answer reduce to for large z (or, equivalently, small R)?