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

on Thursday afternoons (or
sometimes on Fridays).

**2015**

**HW 1, posted Jan. 8, deadline Jan. 15. You only need
to return Problems 1, 2, and 3 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. Read sections I.1, I.2, I.3,
and I.6 of

you will understand the fine
details of the discussion (I certainly don’t). This is just

for your general background
in physics.

5. In the same spirit as in 4.,
just glance at the publications on the mass of the photon in the

Lectures portion of the web
page. Do not spend more than 15-20 minutes on this. Just

enough to understand how
different the world would be if the photons were massive!

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

**HW 2, posted Jan. 15, deadline Jan. 22**

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. You can also consult my notes

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.

2.

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

done
after the lecture this Tuesday January 20.

3.

lecture this Tuesday January 20.

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

**HW 3, posted Jan. 22, deadline Jan. 29**

1.

Green functions that we will have in the homework, thus make

sure
you understand how to deal with these functions. Use

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.

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

5. Show that Eq.(2.19) is
correct, starting with Eq.(2.16).

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

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

**HW 4, posted Jan. 29, deadline Feb. 5**

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.

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

**HW 5, posted Feb. 5, deadline Feb. 12. **

1.

2.

3. Not from

the lecture of next Tuesday.
An electric point dipole of magnitude *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 Feb. 19, deadline extended to March 4 due
to weather delays. **

1.

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

4.

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

or not continuous at the interface.

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

**HW 7, posted March 3, deadline Tuesday March 10. **

** **

1. 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.

2. 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.

3. 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.

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

**HW 8, posted March 5, deadline Friday March 13. **

** **

1. 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.

2. 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”.

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

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

Related to this, watch the
infomercial found here.

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

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

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

**HW 9, posted March 12, deadline March 24 (Tuesday
after Spring Break). **

1. 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.

[After done with

American Journal of Physics 73, 141 (2005) or

American Journal of Physics 68, 1002 (2000)

so that you learn that sometimes problems treated casually
in

or

the solution in

2.

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

3. 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 10, posted March 24, deadline April 6 (delay caused
by**

**second mid term exam)**

1.

2.

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

**HW 11, posted April 7, deadline April 14 **

This homework can be done
only after the lecture of Thursday April 9.

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.

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

**HW 12, posted April 14, deadline April 21.**

1.

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

It is not a centered fed antenna.

2. Solve example 10.3, page
433,

find t_r vs. t, r, and v and then find the explicit scalar

potential V given in Eq.(10.42).

3. From the
“Lectures” portion of this web page corresponding to April 14,
2015,

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

antenna can be made.

4. From the
“Lectures” portion of this web page corresponding to April 14,
2015,

read “Mathematical proof that
1/(1-v/c) ….” to understand how the correction factor

can be deduced mathematically in a rigorous
manner.

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

**HW 13, posted April 16, deadline April 23.**

**Last homework of the semester! Leave directly in
mailbox of grader.**

1. 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 22” I have

included copies of the
relevant portion of

2. This one can be done only
after the lecture of Tuesday April 21

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.

This will be done almost
entirely in class, so it is easy.

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

**Not given this year** :

Glance at

http://en.wikipedia.org/wiki/Gauge_fixing

to have an idea about the non-triviality
of fixing a gauge,

which is often done so
casually in books.

New item in 2015: 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)