**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
of

**2012**

**Week 1, posted Jan. 12, deadline Jan. 19**

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 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

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**Week 2, posted Jan. 20, deadline Jan. 26**

1.
Show that the "Electric Potential" displaying the polar catastrophe
in

the
"unreconstructed state" of the figure of

http://sces.phys.utk.edu/~dagotto/electromag/scanned-lectures/polar.hwang.pdf

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

http://sces.phys.utk.edu/~dagotto/electromag/scanned-lectures/oxide-interfaces.pdf

Basically I am asking to show that the figure on page
6 of my notes is

correct i.e. "no polar catastrophe".

2.

hand. This problem is to remind you of the method of images.

3.

lecture this Tuesday.

4.

Green functions that we will have in the homework, thus make sure you

understand how to deal with these functions.

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**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.**

1.

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. **

1.

2.

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.

1.

2.

3. Starting
with Eq. (5.14) in the book, derive in detail Eq. (5.22).

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**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.

2 .

3.

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
casually in

or

3.

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**Week 9, posted March 15, deadline March 29 (after
Spring break).**

1.

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
book of

to follow this book if you go
line by line through the

math and understand what is being done.

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**Week 10, posted March 29, deadline April 5.**

1.

2.

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**Week 11, posted April 5, deadline April 12.**

1.

2.

3.

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**Week 12, posted April 12, deadline April 24.**

These homework problems are
from

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!!

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