HOMEWORK PROBLEMS. Every week the homework will be posted here on a Thursday (or sometimes Fridays).
Compared to the list of homework problems given in the spring of 2011, the 2012 list will be very similar.
The main difference will be that some of the derivations given in 2011 in class will now be passed to the
homework, to free time at the lectures to cover more chapters
Week 1, posted Jan. 12, deadline Jan. 19
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 calculation.
4. Read sections I.1, I.2, I.3,
and I.6 of
Week 2, posted Jan. 20, deadline Jan. 26
1. Show that the "Electric Potential" displaying the polar catastrophe in
"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.
lecture this Tuesday.
Green functions that we will have in the homework, thus make sure you
understand how to deal with these functions.
Week 3, posted Jan. 26, deadline Feb. 2
1. Show that Eq.(2.5) is correct.
2. Show that Eq.(2.9) is correct. Make a plot of F in F = F (y/y) for a charge Q
of equal sign as the pointlike charge q, and discuss if it changes sign.
Mathematica or other plotting software is fine.
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.
6. Problem 2.23, page 92, only (a) (this is the only “real” problem of
this homework and you can do it only after learning separation of variables).
Week 4, posted Feb. 2, deadline Feb. 9. This homework corresponds to
lectures 6 and 7. From this point on, the lectures will be a bit ahead of
the homework due to the extra lectures on Mondays.
2. 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.
3. Play a bit with the Legendre polynomials and the spherical harmonics. For
instance, 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.
Week 5, posted Feb. 9, deadline Feb. 16. This homework corresponds
to lectures 8 and 9, i.e. still electrostatics.
3. Solve the problem addressed in class sketched in Figure 4.6 but with
the dielectric constant epsilon outside the sphere and vacuum inside, as
in Fig. 4.8. But solve it starting “from scratch” as in Eqs. 4. 48 and 4.49
and eventually arriving to Eqs. 4.59 and 4.60 line by line, as opposed
to exchanging epsilon with epsilon_0.
Week 6, posted Feb. 16, deadline Feb. 23. This homework corresponds
to Magnetostatics, Chapter 5. Although there are cylindrical objects
in the problems, you should not use separation of variables in
cylindrical coordinates (that we skipped) but more elementary
ways to solve the problems.
3. Starting with Eq. (5.14) in the book, derive in detail Eq. (5.22).
Week 7, posted Feb. 23, deadline March 8.
1. Describe briefly what “induction cooking” means and how is
that related with the Faraday’s law.
The important point is to see if Bz and Hz are or not continuous at the interface.
Week 8, posted March 8, deadline March 15.
1. In the context of Section 5.18 (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.
2. 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
Week 9, posted March 15, deadline March 29 (after Spring break).
2. 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 29, deadline April 5.
Week 11, posted April 5, deadline April 12.
Week 12, posted April 12, deadline April 24.
These homework problems are
1. Solve example 10.3 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 in detail. Plot the theta angular dependence of
Eq.(10.68) for the case v/c=0.95.
3. Solve example 11.3 in detail. Plot the theta angular dependence of
Eq.(11.74) for v/c=0.95.
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) becomes 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!!