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 Jackson.

2012

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

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 calculation.

4. Read sections I.1, I.2, I.3, and I.6 of Jackson.

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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. 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.

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.

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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).

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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. Jackson 3.1. No need to do the checks b -> infinity and a -> 0, unless you want to.

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.

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Week 5, posted Feb. 9, deadline Feb. 16.  This homework corresponds

to lectures 8 and 9, i.e. still electrostatics.

1. Jackson 4.1, only part (b).

2. Jackson 4.7, only part (a).

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.

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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. Jackson 5.3.

2. Jackson 5.6.

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 . Jackson 5.13.

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

The important point is to see if Bz and Hz are or not continuous at the interface.

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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

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.

3. Jackson 6.11, part (a) only

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

1. Jackson 6.5, part (a) only.

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 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.

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Week 10, posted March 29, deadline April 5.

1. Jackson 7.2, part (b) only. Two interfaces, have fun!

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

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Week 11, posted April 5, deadline April 12.

1. Jackson 7.4 (a).

2. Jackson 9.3.

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Week 12, posted April 12, deadline April 24.

These homework problems are from Griffiths third edition.

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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