**Problem 1:**

A thin ring of radius R made from a dielectric carries a total charge Q
uniformly distributed over its circumference. It lies in the xy-plane
centered at the origin and is rotating about the z-axis with angular velocity
ω **k**.

(a) Write down a lowest-order expression for **E**(**r**) for r >>
R.

(b) Write down a lowest-order expression for **B**(**r**) for r >>
R.

(c) Does the ring produce electromagnetic radiation? Why or why not?

(d) Is the Pointing vector zero or nonzero for r >> R?

**Problem 2:**

A finite spherically symmetrical charge distribution disperses under
the influence of mutually repulsive forces.

Suppose that the charge
density ρ(r, t), as a function of the distance r from the center of symmetry and
of time, is known.

(a) Prove that the curl of the magnetic field is zero at any point.

(b) Use this to show that **B** is zero at any point.

**Problem 3:**

Assume magnetic charges exist and Maxwell's equations are of the form

**∇**∙**E** = ρ/ε_{0, }**∇**∙**B** = μ_{0}ρ_{m, }-**∇**×**E**
= μ_{0}**j _{m}** + ∂

(a) Write down expressions for the electric field

(b) Write down expressions for the momentum density

(c) Show that there is electromagnetic angular momentum L

**Problem 4:**

Given are Maxwell's equations in macroscopic
form,

**∇**∙**D** = ρ_{f}, **∇**×**E** = -∂**B**/∂t, **
∇**∙**B** =
0, **∇**×**H** = **j**_{f} + ∂**D**/∂t,

and the constitutive relations for a linear, isotropic medium

**D** = ε**E**, **B** = μ**H**.

Let **j**_{f} = σ_{c}**E** + **j**_{b}, where
σ_{c} is the conductivity of the medium and
**j**_{b}
represents the density of forced currents, such as currents inside a
battery.

(a) What is the relationship between the field quantities **E** and
**B**
and the potentials **A** and V?

(b) **E** and
**B** do not specify **A** and V
uniquely. Let
**A** = **A**' - **∇**ψ, V = V' + ∂ψ/∂t. Show that
Maxwell's equations are satisfied by **A**' and V' if they are satisfied
by **A** and V.

(c) Briefly discuss the gauge transformation represented in part b and its
implications.

(d) Write the relationship between the fields given in answers to part
(a) in terms of **H**, **D**, **A**, and V.

(e) Find the wave equation for
**A** in terms of V, μ , ε , **j**_{b},
σ_{c} , and
v = (μ_{0}ε_{0})^{-½} in the Lorentz
gauge.

(f) Starting with Maxwell's equations, derive the wave equation for V in the Lorentz
gauge.

**Problem 5:**

A thin wire of radius b is used to form a circular wire loop of radius a (a
>> b) and total resistance R.

The loop is rotating about the z-axis with constant angular velocity ω**k**
in a region with constant magnetic field **B** = B_{0}**i**.

At t = 0 the loop lies in the y-z plane and the point A at the center of the
wire crosses the y-axis.

Let the (**θ**/θ) direction be tangential to the loop and be equal to the
positive z direction at point A.

Let the (**φ**/φ) direction be tangential to the wire and be equal to the
direction indicated in the figure.

(a) Find the current flowing in the loop. Neglect the
self-inductance of the loop. What is current density **J** as a
function of time?

(b) Find the thermal energy generated per unit time, averaged over one
revolution.

(c) Write down an expression for the the pointing vector **S** on the
surface of the wire.

(d) Use **S** to find the field energy per unit time flowing into the
wire, averaged over one revolution.