Assignment 10

Problem 1:

A dielectric sphere radius "a" and relative permittivity εr 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 ρtrue, a polarization or bound charge density ρbound, and a total charge density ρtotal, such that ρtrue + ρbound = ρtotal, 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 C1 and C2 establishes an electric field in the ground.  One measures the voltage V between a pair of electrodes, with P1 and P2 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(x2 - a2)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 C1 plus another radial distribution converging on C2.

Problem 4:

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 VA' is applied between the two wires at point A, the voltage between the two wires at point B is VB'.  However if a voltage VB" is applied between the two wires at point B, the voltage at A is VA".  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, VA' , VB',  VA" and VB".  Check your answer by showing that x = 19 miles if  L = 50 miles, VA'  = 200 volt, VA" = 40 volt, VB'= 40 volt, VB"  = 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.