**HOMEWORK PROBLEMS.** Every week the homework will be posted here

on Thursday afternoons (or sometimes on Fridays).

**2013**

**Week 1, posted Jan. 11, deadline Jan. 17.**

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 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

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**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

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. It can be

done after the lecture this coming Tuesday.

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. Since you need the method of

images as well, you will also need the lecture of this Tuesday to
solve it.

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**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.

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**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.

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**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.

2.

3.

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**Week 6, posted Feb. 14, deadline Feb. 21. **

This is an abbreviated
homework since

you will be busy with the mid term exam.

1.

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

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**Week 7, posted Feb. 26, deadline March 7. **

1.

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
casually in

or

2.

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
book of

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).**

1.

2.

**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.**

1.

2.

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
from

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!!

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