**Problem 1:**

(a) A positive charge Q is spread over an semicircular arc with radius R as
shown.

(1) What is the work required to bring in a charge -q from infinity to the
center of the arc?

(2) Calculate the magnitude and direction of the force on a charge –q at the
center of the arc.

(b) The potential of a uniformly charged spherical shell of radius R centered
at the origin is

V(r) = q/(4πε_{0}r) r ≥ R, V(r) = q/(4πε_{0}R) r < R,

where q denotes the total charge of the sphere.

Calculate the energy that it requires to deposit a charge Q on an initially
neutral conducting spherical shell with radius R. Use two different approaches
to come to the result.

(3) Calculate the energy by incrementally adding a charge dq to the sphere.

(4) Obtain the energy by considering the resulting electric field of the
spherical shell.

**Problem 2:**

A model of the hydrogen atom was proposed before the advent of quantum
mechanics, which consists of a single electron of mass m and an immobile uniform
spherical distribution of positive charge with radius R. Assume that the
positive charge interacts with the electron via the usual Coulomb interaction
but otherwise does not offer any resistance to the motion of the electron.

(a) Explain why the electron’s equilibrium position is at the center of the
positive charge.

(b) Show that the electron will undergo simple harmonic motion if it is
displaced a distance
d < R away from the center of the positive charge. Calculate its frequency of
oscillation.

(c) How big would the atom need to be in order to emit red light with a
frequency of 4.57*10^{14} Hz? Compare your answer with the radius of the hydrogen
atom.

(d) If the electron is displaced a distance d > R from the center, will it
oscillate in position? Will it undergo simple harmonic motion? Explain!

**Problem 3:**

A charge array consists of two charges, each of magnitude +q, located on
the z-axis at (0, 0, ±a).

(a) Find the potential V(0,0,z) at an arbitrary point z > a; then expand V(0,
0,z) in a power series in z.

(b) Using this series as a “boundary condition”, find the potential V(r,θ,φ) at
an arbitrary location (r,θ,φ) with r > a. An infinite series is
acceptable.

(c) Characterize the first three terms in 1/r^{n}.

**Problem 4:**

Compute the force of attraction between a neutral metallic sphere of radius
a and a point charge q positioned a distance r from the center of the sphere,
where r > a.

How does the force of attraction behave as a function of r if r >> a?

**Problem 5:**

A spherical charge distribution is given by

ρ = ρ_{0}(1 - r/a),
r <
a,

ρ = 0, r > a.

(a) Calculate the total charge Q.

(b) Find the electric field and potential for r > a.

(c) Find the electric field and potential for r < a.

(d) Find the electrostatic energy of this charge
distribution.