HOMEWORK PROBLEMS. Every week the homework will be posted here
on Thursday afternoons (or sometimes on Fridays).
HW 1, posted Jan. 9, deadline Jan. 16. 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 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
the charge takes place.
hand. This problem is to remind you of the method of images. It can be
after the lecture this Tuesday Jan. 21st.
lecture this Tuesday Jan. 21st.
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.21st
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. 21st (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.
3. 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 Feb. 20, deadline Feb. 27.
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.
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:
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
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
4. Glance at
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
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
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 April 3, deadline (by popular demand) April 14 before noon
in the grader’s mailbox.
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.
example of quadrupolar radiation pattern (as opposed to dipolar)
3. 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).
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 x1 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!!