**HOMEWORK PROBLEMS.** Every week the homework will be posted here

on Thursday afternoons (or sometimes on Fridays).

**HW 1, posted Jan. 14, deadline Jan. 21. You only need
to return Problems 1, 2, 3, and 5 below.**

1.

2.

3. Show that Eq. (1.17) (page 35,
top) satisfies the Poisson equation by following

the “a-potential”
procedure described on page 35. This
problem illustrates

how to perform rigorous
mathematical proofs in the E&M context, where the

singular function
1/|x-x’| appears often in the calculations. You will not be requested

to do this rigorous method in
future homework and exams, but it is important that you

learn the method exists.
NOTE: in the eq. below where it says “Direct integration
yields’’

do not worry about showing
that the corrections are O(a^2/R^2) or O(a^2,a^2log(a)). Just

focus on the dominant term, otherwise this
problem will take too long!

4. Glance at the publications on
the mass of the photon in the

Lectures portion of the web
page. As discussed in class, I do not expect that you will

understand the fine details
of the discussion. Thus, do not spend more than ~20 minutes

on this. Just enough to
visualize how different the world would be if photons were massive!

5. On Problem 3.2 of

class). Consider the case of
a cube where at each vertex a charge of value +Q is located.

Calculate the electric
potential at (a) the center of the cube (the naïve “equilibrium
point”)

and at (b) the center of any
of the faces of the cube to show that the cube’s center is not a

minimum for the potential but
there are escape directions along which the potential decreases.

Then just glance at “Quadrupole ion trap” in Wikipedia and learn from the
first figure that

to keep charge approximately
in the same location in space, time dependent fields are needed.

===================================================

**HW 2, posted Jan. 26, deadline Feb. 2**

1.
Show that the "Electric Potential" displaying the polar catastrophe
in

the "unreconstructed state" of the figure of this paper

becomes the potential in the "reconstructed state" in the same figure

after a transfer of charge from the top TiO2 layer inside the structure

to
the top surface layer.

Basically,
I am asking to show that the figure on page 6 of my notes
is

correct i.e. "no polar catastrophe" occurs after the proper
rearrangement

of
the charge takes place (sorry page 2 is upside down in “my notes”).

2. Glance at this publication.
It is a very recent example of polar reconstruction

Here
when charge transfers from top to bottom to avoid the polar catastrophe in

addition
it induces a magnetic phase transition (often insulators are antiferromagnetic

and
sometimes metals are ferromagnets).

3.

hand. This problem is to remind you of the method of images. It can be

done
after the lecture this Thursday January 28.

4.

lecture this Thursday January 28.

===================================================

**HW 3, posted Feb. 2, deadline Feb. 9**

1.

the
method of images to get the Green
function.

2. Show that Eq.(2.5) is
correct, starting with Eqs.(2.3 and 2.4).

3. Show that Eq.(2.9) is
correct, starting with Eq.(2.8)

Plot by hand the force F vs.
distance y, for Q>0, showing

that it is attractive near
the surface and repulsive far away,

as discussed in class Feb 2.

4. Show that Eq.(2.14) is
correct, starting with Eq.(2.12).

5. Show that Eq.(2.19) is
correct, starting with Eq.(2.17) that was

discussed in class.

6. As special case of Eq.(2.19), show that Eq.(2.22) is correct.

===================================================

**HW
4, posted Feb.9, deadline Feb. 16**

1. Problem 2.23, only part (a). Use superposition of solutions.

2.
Confirm that the five P(x)’s in Eq.(3.15) are solutions of Eq.(3.10).

Then,
for l=l’=2 and for l=2, l’=1
confirm that Eq.(3.21) is correct.

Finally,
confirm that Eq.(3.55) is correct for the case l=l’=m=m’=2.

3. Problem 3.1. No need to do
the checks b -> infinity and a -> 0.

4. Show that Eq. (3.36) of
Sec.3.3 is correct, starting with the

general formula Eq. (3.35).
Arriving to the first two terms

of Eq. (3.36) is sufficient.

5.

===================================================

**HW 5, posted Feb. 16, deadline Feb. 19 (yes, you have
less than a week). **

1.

2. Not from *p* pointing along the

*z* axis is at
the center of a sphere of a linear dielectric material (with radius *a* and

dielectric constant epsilon).
Find the scalar electric potential Phi inside and

outside the sphere.

===================================================

**HW 6, posted March 1, deadline March 8 **

1.

2. Starting with Eq. (5.14)
in the book, derive Eq.(5.16)

and then Eq. (5.22).

3. Following the steps in

is one of the results that I
presented without derivation in today’s

lecture.

===================================================

**HW 7, posted March 8, deadline Tuesday March 22. **

** **1.

The important matter to
address is to realize if Bz and Hz are

or not continuous at the interface.

2. Magnetic shielding effect,
section 5.12 of

I want you to go through the
entire calculations on pages 202

and 203 of the book. Start
with the most general magnetic

scalar potentials 5.117 and
5.118. Then write in detail the

boundary conditions 5.119 and
deduce 5.120. Finally, solve

these equations arriving to
5.121 (this part is tedious), and

take the limit of large mu to
arrive to 5.122.

3. From the Wikipedia link
below, read about mu-metals,

namely materials with a high
mu that are useful for

magnetic shielding:

http://en.wikipedia.org/wiki/Mu-metal

In particular, click where it
says “reluctance” in the first line

of magnetic shielding and
learn its meaning. It helps in

understanding why the lines
of magnetic field B tend to

accumulate into the material
with the high mu.

4. This is one of those
“read problems” that allows you

to make contact between what
we are learning and

some recent research efforts:

Read the 2014 Phys. Rev. Lett. paper found here,

on new ways to transfer
magnetic fields over long distances.

I do not pretend at all that
you will understand every line

of the paper (certainly I do
not!) but the goal is simply

for you to get an idea of
current research in E&M.

5. Confirm that the electric
field is indeed given by expression

Eq.(5.167) starting with the
H of 5.166. Then confirm the result

for the “effective
surface current” K_y(t) a few lines below.

Since “delta” is
often small, the current J is confined to be mainly

close to the surface and it
circulates in a manner that reduces

the external magnetic field.

6. Still in Section 5.18 A.
(now with focus on induction

heating) show that the time-averaged power input per unit

volume is given by (5.169),
starting with the expressions

for the current and electric
fields a few lines above. This is

the basis for the concept of
“resistive heating”.

===================================================

**HW 8, posted March 22, deadline Tuesday March 29. **

** **

1. Describe in one page what
“induction cooking” is based on

http://www.explainthatstuff.com/induction-cooktops.html

2. Describe in one page what
“eddy current brake” is based on

http://www.explainthatstuff.com/eddy-current-brakes.html

3. Solve “Example
15” of

or page 348, third edition).
This is about a current flowing

down a wire and the
calculation aims to find the energy per

unit time delivered to the
wire using the Poynting vector.

The problem is solved in

you to do all the math step
by step.

4.

explained in class, but do
all the steps line by line again.

5. Determine the net force on
the “northern” hemisphere of a uniformly charged

solid sphere of radius R and
charge Q, using the Maxwell stress tensor.

This problem is solved in the
book of

to follow this book if you go
line by line through the

math and understand what is
being done.

===================================================

**HW 9, posted March 29, deadline Tuesday April 12 **

**(two weeks because of the second mid term exam)**

1.

2.

===================================================

**HW 10, posted April 12, deadline Tuesday April 19 **

1. Following the steps in the
book, but of course providing more detail,

go from Eq.(9.13) to (9.16).

2. Same as previous problem,
but now starting with (9.16) and ending

with (9.18, top formula for
H). Note: I will have mercy and not ask

you to derive 9.18 bottom
formula for E.

3.

4. Just glance at this link to see a PDF presentation
related with the potential

dangers of cell phone tower
radiation. I myself do not have an opinion on this

subject, but you may find
interesting reading about it. More specifically page 13

contains the radiation
pattern, closely resembling the sin^2theta that was discussed

in the lecture. Page 14 is
the apartment with cancer cases.

5. Glance at this Wikipedia link to explain diffuse sky
radiation and the blue color of the sky.

Inside that article you can
click under Rayleigh scattering as well for more information.

**=========================================**

**HW 11, posted April 19, deadline Tuesday April 26.**

**This is the last homework of the semester!!**

1.

example of quadrupolar radiation pattern (as opposed to dipolar).

It is not a centered fed antenna.

2. From
the “Lectures” portion of this web page corresponding to April 14,
2016,

read “Notes on
designing antennas …” so that you comprehend how a unidirectional

antenna can be made.
It is only 6 pages.

3. Solve example 10.3, page
433,

find t_r as a function of (t,r,v) and
then find the explicit scalar

potential V given
in Eq.(10.42).

4. Starting with equation
(11.66) of

velocity zero and
acceleration nonzero. Then, jump to Example 11.3 and show that (11.74)

is correct for
velocity and acceleration collinear. (if you do not
have

the book, in the
web page of the class in the vicinity of “Lecture 23” I have

included copies of
the relevant portion of

===================================================

**Not given this year :**

Principles of DC generators
can be found here.

Consider the example of page
559

moving at a constant
velocity. Starting with the E’ and B’ fields

shown there, deduce the E and B fields given in (11.152)

Show that the D’Alembertian operator that appears in (11.4)

is invariant under the Lorentz transformation
Eq.(11.16). Use the chain

rule for derivatives, and also use just one spatial
dimension x_{1} for simplicity.