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.
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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.
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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
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.
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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
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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 x1 for simplicity.