Lea: Mathematics for Physicists

Gradshteyn and Ryzhik: Tables of Series, Integrals and Products.

After successfully completing this class, you should be able to:

1. Use basic principles of electromagnetism to analyze physical systems, including systems of conductors, capacitors, wave guides, dielectrics, and current distributions.

2. Bring together ideas from other branches of physics, such as mechanics, thermodynamics, and quantum mechanics, when necessary to understand the behavior of a system.

3. Discuss the principles that apply in a given system, and clearly articulate the solution method.

4. Apply mathematical techniques such as separation of variables in the solution of boundary value problems in electromagnetism in rectangular, spherical and cylindrical geometries.

5. Use the Green's function method to solve Poisson's equation in rectangular, spherical and cylindrical geometries, and to solve the wave equation in free space.

6. Use conservation principles to discuss the evolution of electromagnetic systems.

7. Describe the propagation of electromagnetic waves through differing media and in wave guides.

8. Use computer techniques to solve boundary value problems in electromagnetism.

9. Use computers to plot solutions to problems in electromagnetism.

10. Use computers to evaluate series solutions numerically.

11. Communicate your ideas clearly, orally and in writing.

Physics 704 is to cover the material in the first 8 chapters of Jackson's book; 705 covers the rest! This is a tall order, for you and for me. The objective in studying this material is not just to learn the physics of electromagnetic phenomena, but also to learn a set of mathematical techniques (analytic and numerical) that are useful in many other branches of physics. In order to present the material in a way which emphasizes how a given technique can be applied in differing physical situations, and thus to avoid a good deal of duplication of material, I have attempted to construct a class schedule which does not follow the order of Jackson exactly. It is given below. We may not adhere to this schedule precisely, but it should be a guide for your reading and we can discuss how to proceed as things deviate from the plan. Most of the physics we discuss will not be new to you: you have studied E&M in Phys 230, 360 and 460 (or their equivalents in your own UG institution). We will be taking these concepts deeper and broader, and applying them to a wider range of systems.

Please read the appropriate section of the text **before**
class. The material will undoubtedly be somewhat mysterious the first
time you see
it, and you will get more from each class if you have looked at the
material
in advance. There are also links to my lecture notes on the class schedule below. "Pop" quizzes may happen at any time.

Please have the notes available to you during class (on paper or on a computer) so that I do not have to write all the math on the board.

Doing problems is the essence of a class based on Jackson. Assigned problems are listed on the
schedule. Please check all links as there are additional problems and/or hints listed there. I shall
collect
and grade these problems every week. Do not get behind! The farther behind you
get,
the harder it is to catch up. Here are some
guidelines
for preparing your homework papers.Your grade for the class will be
heavily
based on these problem grades, although a good deal of credit will be
given
for a good attempt. (For what "good" means, check here.) You should include a clear
and concise discussion of relevant physical principles and mathematical
techniques in your solutions, and always analyze your result. Check
this list for things you should NOT say
in your solutions!

I strongly recommend that you plan to spend at least 30 minutes per
week consulting with me in office hours. Check my office hours
now, and
let me know if it is impossible for you to attend any of them.

I do not usually hand out homework solutions. However, you may
obtain a solution if you agree to write a one-paragraph discussion of
the solution, including such factors as how it differs from your own
attempt, and what you have learned from reading it. Such
solutions are for your
own personal use and should not be shared with other students.

There will be a take-home midterm, and an in-class and a take-home final. Problems and the midterm are due at the beginning of the class period on the day indicated. Assignments turned in late will be accepted only under exceptional circumstances.

While I encourage you to discuss the problems during the semester in study groups, please be sure that the work you turn in is your own. Exams may not be discussed with anyone except me. You can probably find solutions to many of the problems on the internet. These solutions range from pretty good to outright wrong. If you can tell the difference, you don't need the solution. If you can't, the solution won't help you. Please review the department's plagiarism policy. Any use of such internet resources is strictly forbidden.

Please note that some of the assignments will involve a computer calculation. You must have some familiarity with at least one computer language such as C++, FORTRAN, BASIC, IDL, PYTHON, or a math package such as MATHEMATICA or MAPLE or MATLAB. Computers may also be (and should be) used to construct plots and diagrams in other assignments.

Grades will be assigned on the following basis:

Class participation | Homework problems: | Midterm: | Final: | Project |

5% | 25% | 30% | 35% | 5% |

If you cannot agree to do the homework problems without external aids, you may choose to have your grades assigned according to the following Plan B scheme:

Homework problems: | Midterm: | Final: | Project |

0 % | 40% | 55% | 5% |

I am a tough grader, so the numbers you get on your assignments may be lower that you have been getting in other classes. An overall score of 80% or better usually gets an A. I do not grade on a curve, because the small numbers in graduate classes make the statistics unreliable. I compare you with the students who have taken this class over the last 20 years.

Please feel free to discuss all aspects of the class with me at any time. Discuss the homework problems among yourselves as well as with me (exams should not be discussed, however). When you discuss problems with other students, general strategies should be discussed, not specific details. For example, you might discuss whether to use a method that minimizes the energy of a system. However, the mathematical details of the method should not be done as a group: that you should do on your own.

Challenge each other! Do not accept what someone else says unless she/he can justify his/her reasoning. Try to attend published office hours, but also feel free to knock on my door whenever I am there. (I'll tell you if I am busy!) It's usually a good idea to make an appointment if you are coming outside office hours.

As graduate students, more is expected of you. You may find it
helpful, indeed necessary, to use reference materials other than
Jackson. You will need a reference that discusses the basic physical
principles: I recommend the Feynman lectures, and also Lea and Burke.
You should have access to
a mathematical reference work listing integrals as well as properties
of
mathematical functions such as Legendre polynomials and Bessel
functions.
The book store has a few copies of Gradshteyn and Ryzhik if you want
your
own: otherwise using the library should suffice. Other books dealing
with
the material include: Lea, *Mathematics for Physicists*,
especially
Chapter 8 and Optional Topic C; Landau and Lifshitz, *Classical
Theory
of Fields*; Schwinger et al, Classical
Electrodynamics, Morse and Feshbach, *Methods of
Theoretical
Physics*; Jeffreys and Jeffreys, *Mathematical Physics*,
especially
Chapters 6, 18, 21, 22 and 24. A good reference for numerical values of
functions
is Ambramowitz and Stegun, *Handbook of Mathematical Functions*
(USGPO
and also Dover). For numerical techniques, I recommend *Numerical
Recipes*,
Press et al.

Pages 31-49 in this newsletter give a good summary of scientific writing dos and don'ts.

This class will be a challenge for all of us, and I hope that we can
meet it together.

Finally, check out another student's take on this class: JacksonForLife.pdf

This is pretty good advice too.

Finally some advice from another professor (courtesy of Dr. Robert Brown at Duke University). Americans with Disabilities Act (ADA) Accommodation: The University is committed to providing reasonable academic accommodation to students with disabilities. The Disability Programs and Resources Center provides university academic support services and specialized assistance to students with disabilities. Individuals with physical, perceptual, or learning disabilities as addressed by the Americans with Disabilities Act should contact Services for Students with Disabilities for information regarding accommodations. Please notify your instructor so that reasonable efforts can be made to accommodate you. If you expect accommodation through the act, you must make a formal request through Disability Programs & Resource Center in SSB110, Telephone 338-2472.Physics 704 | |
Course Outline |
Spring 2020 | |
---|---|---|---|---|

Date | Jackson Reference | Topic (click on links for lecture notes) |
Problems | Due date |

Tu Jan 28 | Introduction (p1-23) Appendices | Overview: Fields and particles. Maxwell's equations.
Units. Boundary conditions Nature of the mathematical problem. Linearity. |
||

Th Jan 30 |
6.1, 1.1-4, 5.1-3, 5.15 | Derivation of Maxwell's equations from experimental results. | 1.1 Addendum to P1.1 |
Jan 30 |

Tu Feb 4 |
1.5-1.7, 5.4, 5.9 6.2, 6.3 , 7.1 |
Scalar and vector potentials. Gauge conditions. Point charge and dipole potentials. Vector and scalar magnetic potentials. | ||

Th Feb 6 |
1.11, 5.16, 6.7,6.8,6.9 | Energy in the EM field.
Capacitance. End of survey of basics. |
1.2, plus additional part, 1.10 What do you get if there is charge inside the sphere? | Feb 6 |

Tu Feb 11 |
1.8-10 |
Effect of boundary conditions. Green's Theorem; uniqueness. Formal theorems | ||

Th Feb 13 | 1.12-13 | Numerical methods | 5.5, 6.13 | Feb 13 |

Fri Feb 14 |
Last day to drop a class | |||

Tu Feb 18 | 2.1-7 |
Derivation
of potential: method of images
Use of images to construct Green function for sphere. |
||

Th Feb 20 | 2.8-2.11 Lea Chapter 8 sections 8.1 & 2 |
Orthogonal functions: Sturm-Liouville problem. Separation of variables. Rectangular 2-D and 3-D potential problems. Fourier integral. | 1.17, 1.20 | Feb 20 |

Tu Feb 25 |
2.8-2.11 Lea Chapter 8 sections 8.1 & 2 2.12 |
Use of conformal transformations in 2-D problems. Polar
coords in 2-D Finite element analysis, Example |
||

Th Feb 27 |
8.1-8.4 |
Boundary conditions on wave solutions: wave guides. | 1.23, 2.4 |
Feb 27 |

Tu Mar 3 |
8.4 |
Rectangular wave guide. | ||

Th Mar 5 | 3.1-3.4 Lea Ch 8 Sec 8.3 |
Separation of variables in spherical coordinates. |
2.26a and b, 2.27, 2.28 |
Mar 5 |

Tu Mar 10 | 3.5 Lea Ch 8 Sec 8.3 |
Separation of variables in spherical coordinates. Spherical harmonics. |
8.5(a), 3.3 | Mar 12 |

Th Mar 12 | 3.6, 5.5 Lea Ch 8 Sec 8.3 |
The addition theorem Magnetic field due to a current loop Midterm handed out |
||

Tu Mar 17 |
3.7-3.8 Lea Ch 8 Sec 8.4 |
Cylindrical coordinates; Bessel functions. |
Midterm due | Mar 19 |

Th Mar 19 |
8.7, 3.13, 5.13 |
Midterm due at beginning of classCylindrical coordinates; Bessel functions Examples. Mixed boundary conditions |
||

Mar 23 - Mar 27 | Spring Break | |||

Tu Mar 31 | 3.9-3.10 Lea Optional Topic C |
Other problems reducible to Laplace's equation. Current flow. | 3.4, 3.12variant | Apr 2 |

Th Apr 2 |
3.11 | Green's functions in terms of orthogonal functions. I. Spherical coordinates |
||

Tu Apr 7 |
Green's function II: Cylindrical coordinates Wronskian |
3.15, 3.16c | Apr 4 | |

Th Apr 9 | 3.12 Lea Optional Topic C |
Green's function III: General method using eigenfunctions | ||

Class project |
3.23,3.24** | May 12 |
||

Tu Apr 14 | 6.1-6.4 (14.1) | Green's function for wave equation: causality | 3.26, 3.27 |
Apr 16 |

Th Apr 16 | 4.1, 4.2 | Multipole expansions | ||

Tu Apr 21 | 4.3-4.7 | Quick survey of dielectrics. | 4.1, 6.20 | Apr 25 |

Th Apr 23 | 4.3-4.7 | More on dielectrics | ||

Tu Apr 28 |
5.6-5.7 | Magnetic moment, Force and torque | 4.7,4.8 | Apr 30 |

Th Apr 30 |
7.1-7.3 | Properties of waves: Polarization | ||

Tu May 5 |
7.1-7.3 | Fresnel formulae | 5.12, 5.30 | May 8 |

Th May 8 |
7.3- 7.7 | Waves in plasmas | ||

Tu May 12 | 7.7-7.11 | Group velocity, pulses
Class project written paper due |
7.3. Do only one of the two polarizations. | May 14 |

Th May 14 | 7.7-7.11 | Last class. Student reports. Take-home final examination handed out. |
||

Th May 21 |
12:30pm - 2:30 pm |
In class Final Examination. |
||

Th May 21 | 12:30 pm | Take-home final examination due |

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Phys 704 is a core course in the MS degree in Physics:

The program learning objectives correlate with the student learning outcomes for this class as follows:

1. Strong and thorough Knowledge and understanding of, and ability to use, essential concepts and methods in physics.

Class outcomes 1, 2, 3, 6 and 7

2. Strong ability to utilize mathematical relationships and methods to describe physical phenomena

Class outcomes 4, 5

3. Ability to solve problems of considerable difficulty in physics by integrating conceptual understanding, quantitative understanding, logical reasoning, and use of mathematical methods

Class outcomes: 1-7

6. Strong ability to communicate knowledge and results to others in written and oral form.

Class outcomes 3 and 11

7. Strong ability to utilize print and electronic resources, computers, and software to gain information and perform calculations.

Class outcomes 8, 9 and 10

8. Ability to work in teams to solve a problem

Class project.

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