**Problem 1:**

A dielectric sphere radius "a" and relative permittivity
ε_{r} is placed in a uniform
electric field. The field region is large so the presence of the sphere does
not perturb the field sources.

(a) Assuming the field inside the sphere to be uniform and the field
outside to be described by the uniform field plus a dipole field centered on the
sphere, find the fields and the dipole moment induced in the sphere by applying
boundary conditions at one point.

(b) What happens if the sphere is conducting?

**Problem 2:**

This question probes your understanding of dielectrics and
associated fields and sources. For this question, dielectric means linear isotropic
homogeneous (lih) dielectric.

(a) If one presumes that there exists a true charge density ρ_{true},
a polarization or bound charge density ρ_{bound}, and a
total charge density ρ_{total}, such that ρ_{true} + ρ_{bound }= ρ_{total}, write the source equations for **D**, **E**,
and **P**. Explain the meaning of these equations.
Briefly address
the question: Which of the fields **D **or **E** might be
considered the more fundamental field? Why? Write the equation(s) describing the
relationships between the three field quantities.

(b) Draw a diagram of a lih dielectric of thickness c between the plates
of a parallel plate capacitor of separation d with a gap of thickness e between
each plate and the dielectric. Presume that there is a constant voltage of V volts
applied to the capacitor at time t = 0 (by, for example, connecting a V volt
battery at time t = 0.) Discuss and draw diagrams of the fields at t = dt for very small dt and the
evolution of the fields to t = ∞.

(c) Repeat part (b) with the lih dielectric replaced by a conductor.
Make sure
that you consider all of the fields, creatively defining **P** in the conductor,
masquerading (for purposes of this question) as a dielectric. Comment on the similarities
and differences between the lih dielectric and the conductor as a dielectric.
(It is
strongly suggested that you do not consider part (c) as meaningless.)

Problem 3:

One can locate resistivity anomalies in the ground as shown in the figure below.

The current I flowing between electrodes C_{1 }and
C_{2 }establishes an electric field in the ground. One measures the voltage
V between a pair of electrodes, with P_{1 }and P_{2 }maintained
at a fixed spacing b. With b << a, V/b is equal to E at the position x. Anomalies in ground conductivity show up in the curve of as a function
of x.

Show that if the substrate conductivity is uniform and equal to σ, then

V/b = 2axI/[πσ(x^{2} - a^{2})^{2}].

The electrodes are of finite size. However, you can perform the
calculation on the assumption that they are infinitely small, disregarding the fact that E
and j would then be infinite at their surfaces.

You can use the principle of superposition as follows.
The current in
the ground is the sum of a radial distribution emanating from C_{1 }plus
another radial distribution converging on C_{2}.

The distance between points A
and B along a telegraph line, consisting of a pair of conducting wires, is L.
There is a single leak between the two wires at a distance x from point A. If a
voltage V_{A}^{'} is applied between the two wires at point A,
the voltage between the two wires at point B is V_{B}^{'}.
However if a voltage V_{B}" is applied between the two wires at point B,
the voltage at A is V_{A}". Assuming the resistance per unit length of
both wires is ρ derive a relationship giving the distance x of the leak as a
function of L, V_{A}^{'} , V_{B}^{'}, V_{A}"
and V_{B}". Check your answer by showing that x = 19 miles if L = 50
miles, V_{A}^{'} = 200 volt, V_{A}" = 40 volt, V_{B}^{'}=
40 volt, V_{B}" = 300 volt.

**Problem 5:**

In the circuit shown below, all three
voltmeters are ideal^{ }and identical. Each resistor has the same given
resistance R.^{ }Voltage V is also given. Find the reading of each^{
}voltmeter.