HOMEWORK PROBLEMS. Every week the homework will be posted here
on Thursday afternoons (or sometimes on Fridays).
Week 1, posted Jan. 11, deadline Jan. 17.
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 carry out 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.
4. Read sections I.1, I.2, I.3, and I.6 of
Week 2, posted Jan. 17, deadline Jan. 24
1. Show that the "Electric Potential" displaying the polar catastrophe in
the "unreconstructed state" of the figure of
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
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".
hand. This problem is to remind you of the method of images. It can be
done after the lecture this coming Tuesday.
lecture this Tuesday.
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 to solve it.
Week 3, posted Jan. 24, deadline Jan. 31
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 at Eq.(2.12).
4. Show that Eq.(2.19) is correct.
5. Show that Eq.(2.22) is correct.
Week 4, posted Jan. 31, deadline Feb. 7
1. Problem 2.23, page 92, only (a). Use superposition of solutions.
2. Confirm that the five P(x)’s in (3.15) are solutions of (3.10). Then, for
l=l’=2 and for l=2, l’=1 confirm that (3.21) is correct. Finally,
confirm that (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.
Week 5, posted Feb. 7, deadline Feb. 14.
1. 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.
Week 6, posted Feb. 14, deadline Feb. 21.
This is an abbreviated homework since
you will be busy with the mid term exam.
2. Starting with Eq. (5.14) in the book, derive in detail Eq. (5.22).
Week 7, posted Feb. 26, deadline March 7.
The important point is to realize if Bz and Hz are or not
continuous at the interface.
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 (5.122).
3. (a) Starting with (5.159) show that the equation (5.160) is correct,
following the steps in the book. (b) Using the boundary conditions
applied to the problem described in Sec. 5.18, item A (the topic we 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.
Week 8, posted March 7, deadline March 14.
1. Search in the internet and describe in one page what
“induction cooking” is.
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. Search in the internet and describe in one page what
“eddy current brake” is.
4. Re-derive Eqs. 6.14, 6.15, and 6.16, i.e. do slowly and properly
what the teacher rushed in the lecture on the subject.
5. Read the first paragraphs of “gauge fixing” to just have an idea
about the non-triviality of fixing a gauge, which is often done
so casually in books.
Week 9, posted March 14, deadline March 21.
1. In class, I explained (or tried to explain) problem “Example 15” of
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. In this context, 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
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.
Week 10, posted March 21, deadline April 4 (week after Spring Break).
Week 11, posted April 4, deadline April 11.
This is about the many formulas of Chapter 9 that I did not
derive explicitly in class today April 4:
1. Arrive to (9.5) using one of the Maxwell equations outside the source
and assuming only one frequency omega.
2. Following the steps in the book, but of course providing more detail,
go from (9.13) to (9.16).
3. Same as 2. but now going from (9.16) to (9.18, top formula).
Note: I will have mercy and not ask you to derive 9.18 (bottom formula)!
4. From both equations (9.18) taken as a given, arrive to both equations
in (9.19) in the “radiation zone” i.e. when r is very large. Eq.(9.19) is
when we show that far from the sources, the fields behave as that of
a plane wave.
Week 12, posted April 11, deadline April 18.
example of quadrupolar radiation pattern
Week 13, posted April 18, deadline April 25.
Last homework of the semester! Leave directly in mailbox of grader.
These homework problems are
1. Solve example 10.3, page 433, in detail i.e. find t_r vs. t, r, and v and then
find the potential V in Eq.(10.42).
2. Solve example 10.4, page 439, in detail. Plot the theta angular
dependence of Eq.(10.68) for the case v/c=0.95.
3. Solve example 11.3, page 463, in detail. Plot the theta angular
dependence of Eq.(11.74) for v/c=0.95.
Now we switch to
4. Show that the wave equation (11.4) in the K’ system transforms to
Eq.(11.5) under a Galilean transformation, i.e. it is not invariant. Use
the chain rule for derivatives. Then show that (11.4) is invariant
under the Lorentz transformation Eq.(11.16). You can use just one spatial
dimension for simplicity.
This was the last homework of the semester!!