HOMEWORK PROBLEMS. Every week the homework will be posted here

on Thursday afternoons (or sometimes on Fridays).

2014

HW 1, posted Jan. 9, deadline Jan. 16. You only need to return Problems 1, 2, and 3 below.

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 perform 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, but it is important that you

learn it exists. NOTE: in the equation right below where it says “Direct integration yields’’

do not worry about showing that the corrections are O(a^2/R^2) or O(a^2,a^2log(a)). Just

focus on the dominant term, otherwise this will take too long!

4. Read sections I.1, I.2, I.3, and I.6 of Jackson. As discussed in class, I do not expect that

you will understand the fine details of the discussion (I certainly don’t). This is just

for your general background in physics.

5. In the same spirit as in 4., just glance at the publications on the mass of the photon in the

Lectures portion of the web page. Do not spend more than 15-20 minutes on this. Just

enough to understand how different the world would be if the photons were massive!

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HW 2, posted Jan. 16, deadline Jan. 23

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

the "unreconstructed state" of the figure of this paper
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 right
before the surface. You can also consult

Basically, I am asking to show that the figure on page 6 of my notes is
correct i.e. "no polar catastrophe" occurs after the proper rearrangement

of the charge takes place.

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 Tuesday Jan. 21st.

3. Jackson 2.2, only (a) and (b). This problem can be done after the
lecture this Tuesday Jan. 21st.

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 Jan.21st

to solve it.

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HW 3, posted Jan. 23, deadline Jan. 30

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 with Eq.(2.12).

4. Show that Eq.(2.19) is correct. Here you have to return to the Green’s

function discussion of the lecture last Tuesday Jan. 21st (or use Eq.(2.16)).

5. As special case, show that Eq.(2.22) is correct.

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HW 4, posted Jan. 30, deadline Feb. 6

1. Problem 2.23, only part (a). Use superposition of solutions.

2. Confirm that the five P(x)’s in Eq.(3.15) are solutions of Eq.(3.10).

Then, for l=l’=2  and for l=2, l’=1 confirm that Eq.(3.21) is correct.

Finally, confirm that Eq.(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.

4. 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.

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HW 5, posted Feb. 6, deadline Feb. 13.

1. Jackson 4.1, only part (b).

2. Jackson 4.7, only part (a).

3. Not from Jackson. An electric point dipole of magnitude p pointing along the

z axis is at the center of a sphere of a linear dielectric material (with radius a and

dielectric constant epsilon). Find the scalar electric potential Phi inside and

outside the sphere.

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HW 6, posted Feb. 20, deadline Feb. 27.

1. Jackson 5.3.

2. Starting with Eq. (5.14) in the book, derive in detail Eq.(5.16)

and then Eq. (5.22).

3. Trivial one, for completeness since I did not discuss it in the lecture:

from Eq.(5.10) find the sign of the forces between wires for parallel

and anti-parallel currents.

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

The important matter to address is to realize if Bz and Hz are

or not  continuous at the interface.

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HW 7, posted Feb. 27, deadline March 6.

In particular, click where it says “reluctance” in the first line

of magnetic shielding and learn its meaning.

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 then (5.122) at large mu.

3. (a) Starting with (5.159) show that the equation (5.160) is correct,

by following the steps in the book.

(b) Using the boundary conditions applied to the problem described

in Sec. 5.18, item A (to be 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.

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HW 8, posted March 6, deadline March 13.

1. Describe in one page what “induction cooking” is based on

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. Describe in one page what “eddy current brake” is based on

4. Glance at

to have an idea about the non-triviality of  fixing a gauge,

which is often done so casually in books.

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HW 9, posted March 13, deadline March 27 (after Spring Break).

1. (a) Solve “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.

The problem is solved in Griffiths obviously, but I want

you to do all the math step by step.

(b) After done with (a), 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.

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HW 10, posted April 3, deadline (by popular demand) April 14 before noon

1. Jackson 7.2, part (b) only.

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

3. Following the steps in the book, but of course providing more detail,

go from (9.13) to (9.16).

4. Same as 3. but now going from (9.16) to (9.18, top formula for H).

Note: I will have mercy and not ask you to derive 9.18 bottom formula for E!

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HW 11, posted April 10, deadline April 21 before noon

1. Jackson 9.3.

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

3. Solve example 10.3, page 433, Griffiths, in detail i.e.

find t_r vs. t, r,  and v and then find the explicit scalar

potential V given in Eq.(10.42).

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HW 12, posted April 17, deadline April 24.

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

1. Show that the D’Alembertian operator that appears in (11.4) Jackson

is invariant under the Lorentz transformation Eq.(11.16). Use the chain

rule for derivatives, and also use just one spatial dimension x1 for simplicity.

Yes, this was almost done entirely in class, so it is easy.

2. The last item of the last lecture of the semester addressed the example

of page 559 Jackson, involving a charge moving at a constant velocity.

Starting with the E’ and B’ fields shown there, deduce the E and B fields

given in (11.152) Jackson.

This was the last homework of the semester!!