## Description

Problem 1. A point charge q is placed at the point r = 0.

(a) By direct calculation show that the divergence of the electric field div E(r) = 0 for

all r 6= 0. What about the divergence at r = 0?

(b) What is the condition div E(r) should satisfy at r = 0 for Gauss’s law to be valid?

Hint. Use Gauss’s law in the integral form.

(3/2 + 3/2 marks)

Problem 2. Suppose the electric field in some region is found to be E = kr3

rˆ, where k is a constant.

(a) Find the charge density.

(b) Find the total charge contained in a sphere of radius R, centered at the origin.

(Do it in two different ways: (1) by integrating the charge density found in part

(a) and (2) by using the integral form of Gauss’s law.)

(1 + 3 marks)

Problem 3. In a certain region of space, the electric field E is uniform, i.e. is constant in both

magnitude and direction.

(a) Use Gauss’s law to argue that this region of space must be electrically neutral, i.e.,

the volume charge density ρ must be zero.

(b) Is the converse true? That is, in a region of space where there is no charge, must

E be uniform? Explain.

(1 + 1 marks)

Problem 4. A point charge Q > 0 is placed at a vertex of a cube as shown in the figure. Find the

electric flux through the surface ABCD.

(4 marks)

Problem 5. A solid insulating ball with radius R is charged non-uniformly: the volume charge

density ρ = Ar/R, where A is a positive constant, and r is the distance from the center

of the ball

(a) Show that the total charge of the ball is Q = πAR3

.

(b) Find the electric field E(r) both inside and outside of the ball. Sketch |E| as a

function of r.

(1 + 3 marks)

Problem 6. An infinite plane slab of thickness 2d (see the figure below) carries a uniform volume

density ρ. Find the electric field as a function of y, where y = 0 at the center of the

slab. Plot E versus y, calling E positive if it points in the +y direction and negative if

it points in the −y direction.

(5 marks)

Problem 7. Consider an infinite solid insulating cylinder with radius R uniform

charged with constant density ρ < 0. Imagine that a cylindrical hole
with radius a has been bored along the entire length of the cylinder. The
axis of the cavity is a distance b from the axis of the cylinder, where
a < b < R.
Find the magnitude and the direction of the electric field E inside the
cavity to show that E is uniform over the entire cavity.
(4 marks)
Problem 8. Two spherical cavities, of radii ra and rb, are hollowed out from the interior
of a neutral conducting ball of radius R. At the center of each cavity a
point charge is placed: qa and qb, respectively.
(a) Find the surface densities of charge σa, σb on the walls of the cavities
as well as on the surface of the ball σR.
(b) What is the electric field outside of the conductor?
(c) What is the electric field within each cavity?
(d) What is the force on qa and qb?
(e) Which of these answers would change if a third charge qc were
brought near the conductor?
Explain your answers.
(5 × 2 marks)