# Electromagnetic Fields Homework Solutions

### PHY 6346 — Electromagnetic Theory 1 — Fall 2015

**Evaluations**

**Course:** PHY 6346, Electromagnetic Theory 1, **Meeting times:** MWF, Period 8 (3:00-3:50pm), room 1101 **Instructor:** J. N. Fry, **Office:** 2172, **Phone:** 392-6692, **e-mail:**, **Office Hours:** M 9, Tu/Th 6 [schedule]**Graders:** Xiao Chen, **Office:** B51, **Phone:** 392-0477, **e-mail:**, **Office Hour:** W 5

Luyi Yan, **Office:** 1133, **Phone:** 273-4643, **e-mail:**, **Office Hour:** W 5

**Course Description:** PHY 6346 is the first semester of the graduate core sequence in Electromagnetism, studying Maxwell's equations in general and in specific situations. The objectives of the course are (*i*) to study electrodynamics at a theoretically sophisticated level; (*ii*) to develop mathematical techniques useful for solving problems in E&M as well as other areas of physics; (*iii*) to develop problem solving skills; (*iv*) to prepare the student (if necessary) for the preliminary exam. Topics to be covered include

- electrostatics: Coulomb's Law, the electric field, Gauss's Law
- scalar potential, Poisson's equation, Green's functions
- methods: images, separation of variables, harmonic functions
- multipoles, polarization, displacement, linear dielectrics
- magnetostatics: Biot-Savart Law, the magnetic field, Ampère's Law, vector potential, magnetization, hard and linear magnetic materials
- time varying magnetic field: Faraday's Law, induction, Ohm's law, magnetic diffusion, skin depth
- full time-dependent Maxwell equations
- conservation laws, energy density, energy flux (Poynting vector), momentum density, momentum flux (Maxwell stress tensor)
- harmonically varying fields, time average

**Grading:** Grading will be based 50% on periodic homework sets, and 25% each on a midterm and final exam. The midterm exam will be on or about Thursday, October 15, 8:00–10:00 pm (7:30?), in room NEB 202. **[solution]****[distribution]**. The final exam is **Thursday, December 17, 5:30–7:30pm** (Exam Group 17E). some sample problems**[solution]****[distribution]** [average = 56 ± 11, median = 55, quartiles = 49, 63]. Students are expected to complete work at the time it is due, or as soon as possible in case of illness or other accepted, documented unusual circumstance. Homework solutions will be posted as soon as all papers are turned in, and it is not fair to your colleagues to delay this. In extreme cases I will post the names of people who are holding up the availability of solutions. There will be no last-minute makeups accepted.

The following paragraphs of advice on how to do well in Physics are lifted from one of my colleagues. This is a graduate course, and you are free to make your own choices, but you should think about what they say:

I do not plan to take daily attendance, but it is to your advantage to *attend class*. You may spend most of your time sleeping or texting, but in between you will have the opportunity to learn what subjects I think are important, and you can then concentrate on these subjects during your reading. If by some unfortunate set of circumstances you do miss class, do not ask me if I said anything important — everything that I say is important. Instead, ask a classmate; she or he is likely to give an honest answer, and you won't offend me. There will be a substantial number of examples discussed in class that are not in the textbook, and examples in class often appear on tests. If you miss class you will not do well in this course.

*Do the assigned homework.* This is the drudge part of physics, but it is absolutely necessary. We will learn grand ideas and see their wondrous applications in class. But, your understanding is only superficial unless you can apply these same grand ideas to completely new circumstances. In course work, this is usually done with homework problems. Do not be surprised if the homework is frustrating at times; solving one challenging problem makes the next much easier. And homework problems often appear on tests. Doing all of the homework is the easiest way to improve your grade. Not doing homework is the easiest way to lower your grade.

homework-exam correlation and another

**Required text:**

**Other books:**

- Andrew Zangwill,
*Modern Electrodynamics*(Cambridge, 2013). [errata] - David J. Griffiths,
*Introduction to Electromagnetics*, 3/E (Addison-Wesley, 1999), 4/E (2013) - L. D. Landau and E. M. Lifshitz,
*The Classical Theory of Fields*, Fourth Revised English Edition.

Course of Theoretical Physics Volume 2 (Pergamon, 1975, 1987, 1997).^{[1]} - L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii,
*Electrodynamics of Continuous Media*, 2d edition.

Course of Theoretical Physics Volume 8 (Pergamon, 1960, 1984, 1993).^{[1]} - Francis E. Low,
*Classical Field Theory: Electromagnetism and Gravitation*(Wiley, 1997). - Mark A. Heald and Jerry B. Marion,
*Classical Electromagnetic Radiation*, 3rd edition (Brooks Cole, 1994).

(J. B. Marion, 1st ed, Academic Press, 1965).^{[1]}Dover reprint - W. K. H. Panofsky and M. Phillips,
*Classical Electricity and Magnetism*, 2nd edition (Addison-Wesley, 1962). - Edward Purcell,
*Electricity and Magnetism*(McGraw-Hill, 1984).^{[1]} - Julian Schwinger, Lester L. DeRaad, Jr., Kimball A. Milton, and Wu-yang Tsai,
*Classical Electrodynamics*(Perseus, 1998).^{[1]}

“The teaching of electromagnetic theory is something like that of American History in school: you get it again and again. Well, this is the end of the line.” - Davison Eugene Soper,
*Classical Field Theory*(John Wiley & Sons, 1976).^{[2]}^{[1]}Gaussian units;^{[2]}Lorentz units - George B. Arfken, Hans J. Weber, Frank E. Harris,
*Mathematical Methods for Physicists: A comprehensive Guide*, Seventh Edition (Elsevier, 2013). - M. L. Boas,
*Mathematical Methods in the Physical Sciences*, 3rd edition (Wiley 2006) - Philip M. Morse and Herman Feshbach,
*Methods of Theoretical Physics, Parts I, II*(McGraw-Hill, 1953). - Harry M. Schey,
*Div, Grad, Curl, and All That: An Informal Text on Vector Calculus*, Third Edition (W.W. Norton, 1997).

Kevin Schmidt's *Possibly Useful Books for Classical Electromagnetism*, with comments

The following quote (attributed to Sidney Coleman) captures a common reaction to Jackson: Scientists tend to overcompress, to make their arguments difficult to follow by leaving out too many steps. They do this because they have a hard time writing and they would like to get it over with as soon as possible.... Six weeks of work are subsumed into the word “obviously.”

**This and that:**

Physical Constants from the Particle Data Group

JavaScript Calculator with built-in SI and cgs constants [1996 values]

Math trivia, Akira Hirose Math Notes

Charles Augustin de Coulomb, *Accuracy of Coulomb's law*, John Robison

More than you wanted to know about *δ*-functions from

random vector field, longitudinal vector field, transverse vector field

Renormalized potential for an infinite line charge. See also arXiv:physics/0503107 (in Portuguese) *George Green, Mathematician* (Nottinghamshire History), *George Green, Mathematician and Physicist* (*The Mathematical Gazette*),

*The Green of Green Functions* (*Physics Today*, December 2003), From Darkness to Green** An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism** (1828) [] [google books]

*Proof that the Neumann Green's function in electrostatics can be symmetrized*: K. J. Kim and J. D. Jackson, Am. J. Phys. 61, 1144 (1993)

images in parallel mirrors(closeup)(image)(image)

Potential at the center of a cube, 2

*d*frame

Fourier sin series for step functionanimation (135) animation (pbs)

Fourier sin series for

*δ*(

*x*−

*x*') (max = 8, 32, 128, 1024), zoom

Green's function for 2

*d*square

Harmonic functions

Legendre polynomials: Rodrigues' Formula and recursions

*Pl*(cos

*θ*) vs.

*θ*and vs. (

*l*+½)

*θ*

Legendre polynomial expansion of step function

Spherical Harmonic Angular Distributions

Spherical Harmonics

Bessel functions,

*P*

_{100}(cos

*θ*), zeroes

*x*

_{0n},

*x*

_{1n}of

*J*

_{0}(

*x*),

*J*

_{1}(

*x*) Lecture −6 by Dr P B Agarwal (IIT Roorkee)

Bessel series expansion of step function

More about Bessel functions

NASA GSFC and NIMA Joint Geopotential Model, NGA/NASA Earth Gravitational Model, multipole amplitudes

André Marie Ampère

Magnetic Monopoles Song

Magnetic Monopole Searches

*Evidence for Detection of a Moving Magnetic Monopole*, B. Price, E. Shirk, W. Osborne, L. Pinsky,

*Phys Rev Lett*

**35**, 487 (1975) /

*Further measurements and reassment*

*Phys Rev Lett*

**D18**, 1382 (1978).

*First Results from a Superconductive Detector for Moving Magnetic Monopoles*, B. Cabrera,

*Phys Rev Lett*

**48**, 1378 (1982).

Elliptic integrals

Non-dipolar order

Magnetic shielding, wikipedia, MμMetal, lessEMF

cylindrical bar magnet

Michael Faraday, Joseph Henry

Magnetic field diffusion (in) [log

*t*/

*τ*= −5, 5, 1], (out), skin depth, ELF submarine communication

Who was James Clerk Maxwell?,

*A Dynamical Theory of the Electromagnetic Field*, J. Clerk Maxwell,

*Phil. Trans. R. Soc. Lond.*

**155**, 459 (1865).

John Henry Poynting, radiation pressure,

*A*COMPARISON

*of the*FLUCTUATIONS

*in the*PRICE

*of*WHEAT

*and in the*COTTON

*and*SILK IMPORTS

*into*GREATK BRITAIN,

*Journal of the Royal Statistical Society*

**47**, 33–64 (1884)

~~Cosmic Variance~~, @dalcantonJD, @RisaWechsler, sean carroll, @seanmcarroll, Leaves on the Line, @defjaf, Cosmic Yarns, @ktfreese, In the Dark, @telescoper, @lbaudis, Quantum Frontiers, @preskill, @LKrauss1, @FrankWilczek, The Insoluble Pancake, Starts With a Bang!, @StartsWithABang!

Skeetobite Weather, SFWMD computer tracks, HWRF

Lego LHC, Collider, Mr Higgs Twist, [Les Horribles Cernettes]

LHC Street View

Bohemian Gravity, I Will Derive

Let it SnowLet it Snow(Dean Martin), High Tide Ventures Christmas, The Twelve Days of Christmas

**Class Diary**

**Homework:** Problem solving is a skill learned only through practice. Take advantage of the homework as an opportunity to learn how to recognize the right approach to a problem before exam time and it becomes too late. While I encourage you to discuss the assignments with each other, what you turn in must represent your own work. As we also do when publishing research articles: if you obtain significant information from a published or human source, *cite that source*. This will often be as little as "Jackson, eq. (10.43)". If you work together, please identify other members of your working group. This edition of the textbook uses SI (mks) units, at least until the chapters on relativity. Correct solutions to the assigned problems will be in appropriate units. Best answers are reduced to simplest terms; ½ tan^{−1}[2*az*/(*a*^{2}−*z*^{2})] is and is not the same as tan^{−1}(*z*/*a*) . And, please be sure that in the end your elegant solution in fact answers the question asked! Three Tips for Solving Physics Problems / How to Solve Physics Problems

**Evaluations**

**University Policies:** Students are expected to know and comply with the University's policies regarding academic honesty and use of copyrighted materials. Cheating, plagiarism, or other violations of the Academic Honesty Guidelines will not be tolerated and will be pursued through the University's adjudication procedures.

*“Students requesting classroom accommodations must first register with the Disabilities Resources Program, located in the Dean of Students Office, P202 Peabody Hall. The Disabilities Resources Program will provide documentation to the student, who must then deliver this documentation to the instructor when requesting accommodations.”*

EMF called electromagnetic field and practically we can see an example of our earth which is surrounded by electromagnetic field. In electromagnetic field, fundamental law is Maxwellian principals which explain electromagnetic properties of matter, matters which having macroscopic properties explain electromagnetic field theory between source and its interaction with matter. Also here other fundamental topics including electromagnetic field theory, as for example consititutive parameter and boundary condition, we can apply the energy conservation law to bounded volume.

Here common applications low-frequency nonionizing radiation fields and mdash ;those fields below 300 Hertz (Hz)&mdash ; and maximum time only to those fields in which the range is 50 to 60 Hz, which is basically known as power-frequency EMFs. In electromagnetic field theory we have to discuss many law's such as coulomb's law, Faraday's law, Maxwell's law, Ampere-circuital law etc.

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