HOMEWORK PROBLEMS. Every week the homework will be posted here
on Thursday afternoons (or sometimes on Fridays).
HW 1, posted Jan. 8, deadline Jan. 15. You only need to return Problems 1, 2, and 3 below.
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
"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
the charge takes place.
hand. This problem is to remind you of the method of images. It can be
after the lecture this Tuesday January 20.
lecture this Tuesday January 20.
HW 3, posted Jan. 22, deadline Jan. 29
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.
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.
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
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
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
Related to this, watch the infomercial found here.
4. Describe in one page what “eddy current brake” is based on
HW 9, posted March 12, deadline March 24 (Tuesday after Spring Break).
1. Solve “Example
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
the solution in
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
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)
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!
HW 12, posted April 14, deadline April 21.
example of quadrupolar radiation pattern (as opposed to dipolar).
It is not a centered fed antenna.
2. Solve example 10.3, page
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
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 x1 for simplicity.
This will be done almost entirely in class, so it is easy.
Not given this year :
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
moving at a constant velocity. Starting with the E’ and B’ fields
shown there, deduce the E and
B fields given in (11.152)