HOMEWORK PROBLEMS. Every week the homework will be posted here

on Thursday afternoons (or sometimes on Fridays).

 

HW 1, posted Jan. 14, deadline Jan. 21. You only need to return Problems 1, 2, 3, and 5 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. Glance at the publications on the mass of the photon in the

Lectures portion of the web page. As discussed in class, I do not expect that you will

understand the fine details of the discussion. Thus, do not spend more than ~20 minutes

on this. Just enough to visualize how different the world would be if photons were massive!

5. On Problem 3.2 of Griffiths, the Earnshaw theorem is expressed (also briefly mentioned in

class). Consider the case of a cube where at each vertex a charge of value +Q is located.

Calculate the electric potential at (a) the center of the cube (the nave “equilibrium point”)

and at (b) the center of any of the faces of the cube to show that the cube’s center is not a

minimum for the potential but there are escape directions along which the potential decreases.

Then just glance at “Quadrupole ion trap” in Wikipedia and learn from the first figure that

to keep charge approximately in the same location in space, time dependent fields are needed.

 

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HW 2, posted Jan. 26, deadline Feb. 2

 

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.

 

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 (sorry page 2 is upside down in “my notes”).

2. Glance at this publication. It is a very recent example of polar reconstruction

Here when charge transfers from top to bottom to avoid the polar catastrophe in

addition it induces a magnetic phase transition (often insulators are antiferromagnetic

and sometimes metals are ferromagnets).

3. 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 Thursday January 28.

4. Jackson 2.2, only (a) and (b). This problem can be done after the
lecture this Thursday January 28.

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HW 3, posted Feb. 2, deadline Feb. 9

 

1. Jackson 2.7, only (a) and (b). 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 Feb 2.

 

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

 

5. Show that Eq.(2.19) is correct, starting with Eq.(2.17) that was

discussed in class.

 

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

 

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HW 4, posted Feb.9, deadline Feb. 16


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.

5. Jackson 4.1, only part (b).

 

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HW 5, posted Feb. 16, deadline Feb. 19 (yes, you have less than a week).

 

1. Jackson 4.7, only part (a).

2. 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 March 1, deadline March 8

 

1. Jackson 5.3. Use Biot-Savart law.

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

 

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HW 7, posted March 8, deadline Tuesday March 22.

 

1. 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.

 

2. Magnetic shielding effect, section 5.12 of Jackson.

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

and 203 of the book. Start with the most general magnetic

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.

 

3. From the Wikipedia link below, read about mu-metals,

namely materials with a high mu that are useful for

magnetic shielding:

http://en.wikipedia.org/wiki/Mu-metal

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.

 

4. 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.

 

5. 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.

 

6. 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”.

 

 

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HW 8, posted March 22, deadline Tuesday March 29.

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

http://www.explainthatstuff.com/induction-cooktops.html

 

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

http://www.explainthatstuff.com/eddy-current-brakes.html

 

3. 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.

 

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

explained in class, but do all the steps line by line again.

 

5. 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 9, posted March 29, deadline Tuesday April 12

(two weeks because of the 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 10, posted April 12, deadline Tuesday April 19

 

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.

4. Just glance at this link to see a PDF presentation related with the potential

dangers of cell phone tower radiation. I myself do not have an opinion on this

subject, but you may find interesting reading about it. More specifically page 13

contains the radiation pattern, closely resembling the sin^2theta that was discussed

in the lecture. Page 14 is the apartment with cancer cases.

5. Glance at this Wikipedia link to explain diffuse sky radiation and the blue color of the sky.

Inside that article you can click under Rayleigh scattering as well for more information.

 

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HW 11, posted April 19, deadline Tuesday April 26.

This is the last homework of the semester!!

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

example of quadrupolar radiation pattern (as opposed to dipolar).

It is not a centered fed antenna.

2. From the “Lectures” portion of this web page corresponding to April 14, 2016,

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

antenna can be made. It is only 6 pages.

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

find t_r as a function of (t,r,v) and then find the explicit scalar

potential V given in Eq.(10.42).

4. 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 23” I have

included copies of the relevant portion of Griffiths).

 

 

 

 

 

 

 

 

 

 

 

 

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Not given this year :

 

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.

 

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.