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

on Thursday afternoons (or sometimes on Fridays).

**2014**

**HW 1, posted Jan. 9, deadline Jan. 16. 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 it exists. NOTE: in the
equation right 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 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. 16, deadline Jan. 23**

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 to the top layer right

before the surface. 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 Jan. 21st.

3.

lecture this Tuesday Jan. 21st.

4.

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

understand how to deal with these functions. Since you need the method of

images
as well, you will also need the lecture of this Tuesday Jan.21^{st}

to
solve it.

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

**HW 3, posted Jan. 23, deadline Jan. 30**

1. Show that Eq.(2.5) is correct.

2. Show that Eq.(2.9) is
correct. Make a plot, by hand if you wish, of

the force F vs. distance y,
for Q>0, showing that it is attractive near

the surface and repulsive far
away.

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

4. Show that Eq.(2.19) is
correct. Here you have to return to the Green’s

function discussion of the
lecture last Tuesday Jan. 21^{st} (or use Eq.(2.16)).

5. As special case, show that Eq.(2.22) is correct.

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

**HW 4, posted Jan. 30, deadline Feb. 6**

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. 6, deadline Feb. 13. **

1.

2.

3. 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 Feb. 20, deadline Feb. 27. **

1.

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

and then Eq. (5.22).

3. Trivial one, for
completeness since I did not discuss it in the lecture:

from Eq.(5.10) find the sign
of the forces between wires for parallel

and anti-parallel currents.

4.

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

or not continuous at the interface.

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

**HW 7, posted Feb. 27, deadline March 6. **

1. From a Wikipedia link,
read about mu-metals:

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.

2. This is the boring part of
the magnetic shielding calculation:

From (5.119) find the
equations satisfied by the unknown

coefficients, namely arrive
to (5.120), and then solve and arrive

to (5.121) and then (5.122)
at large mu.

3. (a) Starting with (5.159)
show that the equation (5.160) is correct,

by following the steps in the
book.

(b) Using the boundary conditions applied to
the problem described

in Sec. 5.18, item A (to be
discussed in class) show that the magnetic

field has only an x component
at z=0+, if at z=0- it only has an x component.

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

**HW 8, posted March 6, deadline March 13. **

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

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

2. In the context of Section
5.18 A. (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.

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

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

4. 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.

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

**HW 9, posted March 13, deadline March 27 (after Spring
Break). **

1. (a) 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.

(b) After done with (a),
simply glance at the publications

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

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 April 3, deadline (by popular demand)
April 14 before noon**

**in the grader’s mailbox.**

1.

2.

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

go from (9.13) to (9.16).

4. Same as 3. but now going
from (9.16) to (9.18, top formula for H).

Note: I will have mercy and
not ask you to derive 9.18 bottom formula for E!

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

**HW 11, posted April 10, deadline April 21 before noon**

**in the grader’s mailbox.**

1.

2.

example of quadrupolar
radiation pattern (as opposed to dipolar)

3. 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).

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

**HW 12, posted April 17, deadline April 24.**

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

1. 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.

Yes, this was almost done
entirely in class, so it is easy.

2. The last item of the last
lecture of the semester addressed the example

of page 559

Starting with the E’
and B’ fields shown there, deduce the E and B fields

given in (11.152)

This was the last homework of
the semester!!