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.
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.
I am asking to show that the figure on page 6 of my notes
correct i.e. "no polar catastrophe" occurs after the proper rearrangement
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
sometimes metals are ferromagnets).
hand. This problem is to remind you of the method of images. It can be
after the lecture this Thursday January 28.
lecture this Thursday January 28.
HW 3, posted Feb. 2, deadline Feb. 9
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.
HW 5, posted Feb. 16, deadline Feb. 19 (yes, you have less than a week).
2. Not from
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
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
HW 7, posted March 8, deadline Tuesday March 22.
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
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
2. Describe in one page what “eddy current brake” is based on
3. 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.
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
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)
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.
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!!
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
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
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
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 x1 for simplicity.