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. Jackson 1.4

2. Jackson 1.6, only (a), (b), and (c).

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 Jackson.

=================================================

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. Jackson 2.1, only (a), (b), (c), and (d). In (a), just make a plot by
hand. This problem is to remind you of the method of images. It can be

done after the lecture this coming Tuesday.

3. Jackson 2.2, only (a) and (b). This problem can be done after the
lecture this Tuesday.

4. Jackson 2.7, only (a) and (b). This is the only problem about
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.

2. Jackson 4.1, only part (b).

3. Jackson 4.7, only part (a).

===================================================

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

This is an abbreviated homework since

you will be busy with the mid term exam.

1. Jackson 5.3.

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

===================================================

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

1. Jackson 5.19. In (b), plot just by hand for a generic L/a.

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

Griffiths (page 324, second edition; 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.  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 Griffiths

or Jackson are much more difficult than these authors imply.

2. Jackson 6.11, part (a) only. Yes, this problem was partially

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 Griffiths and it is quite fine

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. Jackson 7.2, part (b) only.

2. Jackson 7.3 (a). Note: do electric field perpendicular to “plane of incidence” only.

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. Jackson 9.3.

2. Jackson 9.16 (a). Note: this is an interesting

Week 13, posted April 18, deadline April 25.

Last homework of the semester! Leave directly in mailbox of grader.

These homework problems are from Griffiths third edition.

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 Jackson:

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!!

==============================================================