HOMEWORK PROBLEMS. Every week the homework will be posted here

on Thursday afternoons (or sometimes on Fridays).

2015

HW 1, posted Jan. 8, deadline Jan. 15. 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 the method exists. NOTE: in the eq. 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 problem 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. 15, deadline Jan. 22

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 inside the structure

to the top surface layer. 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 January 20.

3. Jackson 2.2, only (a) and (b). This problem can be done after the
lecture this Tuesday January 20.

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HW 3, posted Jan. 22, deadline Jan. 29

1. 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. Use

the method of images to  get the Green function.

2. Show that Eq.(2.5) is correct, starting with Eqs.(2.3 and 2.4).

3. Show that Eq.(2.9) is correct, starting with Eq.(2.8)

Plot by hand the force F vs. distance y, for Q>0, showing

that it is attractive near the surface and repulsive far away,

as discussed in class.

4. Show that Eq.(2.14) is correct, starting with Eq.(2.12).

5. Show that Eq.(2.19) is correct, starting with Eq.(2.16).

6. As special case of Eq.(2.19), show that Eq.(2.22) is correct.

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HW 4, posted Jan. 29, deadline Feb. 5

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. 5, deadline Feb. 12.

1. Jackson 4.1, only part (b).

2. Jackson 4.7, only part (a).

3. Not from Jackson. This problem you will only be able to do after

the lecture of next Tuesday. 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. 19, deadline extended to March 4 due to weather delays.

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. Following the steps in Jackson, go from Eq.(5.51) to (5.55). This

is one of the results that I presented without derivation in today’s

lecture.

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 March 3, deadline Tuesday March 10.

1. Magnetic shielding effect, section 5.12 of Jackson.

I want you to go through the entire calculations on pages 202

scalar potentials 5.117 and 5.118. Then write in detail the

boundary conditions 5.119 and deduce 5.120. Finally, solve

these equations arriving to 5.121 (this part is tedious), and

take the limit of large mu to arrive to 5.122.

namely materials with a high mu that are useful for

magnetic shielding:

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

of magnetic shielding and learn its meaning. It helps in

understanding why the lines of magnetic field B tend to

accumulate into the material with the high mu.

3. This is one of those “read problems” that allows you

to make contact between what we are learning and

some recent research efforts:

Read the 2014 Phys. Rev. Lett. paper found here,

on new ways to transfer magnetic fields over long distances.

I do not pretend at all that you will understand every line

of the paper (certainly I do not!) but the goal is simply

for you to get an idea of current research in E&M.

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HW 8, posted March 5, deadline Friday March 13.

1. Confirm that the electric field is indeed given by expression

Eq.(5.167) starting with the H of 5.166. Then confirm the result

for the “effective surface current” K_y(t) a few lines below.

Since “delta” is often small, the current J is confined to be mainly

close to the surface and it circulates in a manner that reduces

the external magnetic field.

2. Still in Section 5.18 A. (now with focus on 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. This is

the basis for the concept of “resistive heating”.

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

Related to this, watch the infomercial found here.

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

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HW 9, posted March 12, deadline March 24 (Tuesday after Spring Break).

1. 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.

[After done with Griffiths, 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 more difficult than these authors imply. But for our purposes

the solution in Griffiths is fine].

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

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 March 24, deadline April 6 (delay caused by

second mid term exam)

1. Jackson 7.2, part (b) only.

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

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HW 11, posted April 7, deadline April 14

This homework can be done only after the lecture of Thursday April 9.

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

go from Eq.(9.13) to (9.16).

2. Same as previous problem, but now starting with (9.16) and ending

with (9.18, top formula for H).  Note: I will have mercy and not ask

you to derive 9.18 bottom formula for E!

3. Jackson 9.3.

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HW 12, posted April 14, deadline April 21.

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

It is not a centered fed antenna.

2. 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).

3. From the “Lectures” portion of this web page corresponding to April 14, 2015,

read  “Notes on designing antennas …” so that you comprehend how a unidirectional

4. From the “Lectures” portion of this web page corresponding to April 14, 2015,

read “Mathematical proof that 1/(1-v/c) ….” to understand how the correction factor

can be deduced mathematically in a rigorous manner.

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HW 13, posted April 16, deadline April 23.

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

1. Starting with equation (11.66) of Griffiths third edition  show that (11.69) is correct for

velocity zero and acceleration nonzero. Then, jump to Example 11.3 and show that (11.74)

is correct for velocity and acceleration collinear. (if you do not have

the book, in the web page of the class in the vicinity of “Lecture 22”  I have

included copies of the relevant portion of Griffiths).

2. This one can be done only after  the lecture of Tuesday April 21

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.

This will be done almost entirely in class, so it is easy.

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Not given this year :

Glance at

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

which is often done so casually in books.

New item in 2015: principles of DC generators can be found here.

Consider 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.