Description
Problem 1. Four circular plastic rods are charged uniformly, each with charge Q < 0. Rank the
four arrangements according to the magnitude of the electric field at the origin. Explain
your answer.
(3 marks)
Problem 2. Three charges are placed in the vertices of a square. Find the electric field at points
A, B, and C (the center of the square). For numerical calculations assume: q = 1 nC,
a = 10−10 m.
(3 × 1 marks)
Problem 3. Electric quadrupole consists of two positive and two negative charges of the same absolute value q, arranged as in the figure below. Find the electric field of the quadrupole
at distance r a from its center O, with the line OA parallel to one of the sides of the
square.
(4 marks)
Problem 4. A rod of length l is charged with linear density λ < 0. Find the electric field on the axis of the rod, at distance a from one of the ends, if (a) λ = const, (b) λ = Ax, where A is a constant. (2 + 3 marks) Problem 5. A thin disk with a circular hole in its center has its inner radius R1, and the outer radius R2. The disk is uniformly charged, with surface charge density σ > 0.
(a) Find the electric field on the axis of the symmetry of the disk perpendicular to its
surface.
Rather than solving this problem by direct integration, use results and formulas
derived in the lecture.
(b) Show that at points on this axis, sufficiently close to the geometric center of the
disk, the magnitude of the electric field is approximately proportional to the distance from the center. Then consider a particle with mass m and negative charge
−q, which is placed on this axis, at distance 0.01R1 from the center of the disk and
is free to move along the axis. Show that, after the particle is released at t = 0,
its motion may be treated as harmonic. Find the period of oscillations.
(2 + 3 marks)
Problem 6. Two thin rods of length l lie along the x-axis, one between x = a/2 and x = a/2 + l,
and the other between x = −a/2 and x = −a/2 − l. Each rod has positive charge Q
distributed uniformly along its length.
(a) Show that the magnitude of the force that one rod exerts on the other is
F =
Q2
4πε0l
2
ln ”
(a + l)
2
a(a + 2l)
#
.
(b) Show that if a l, the magnitude of this force reduces to F = Q2/4πε0a
2
. What
is the interpretation of this result?
Hint. For |u| 1, you may find the expansion ln(1 + u) = u − u
2/2 + u
3/3 − . . .
helpful.
(4 + 2 marks)
