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Physics 214 Home Page---Winter 2017

This page contains copies of the class handouts, and other pertinent items of interest for the U.C. Santa Cruz Physics 214 graduate Electromagnetism II class.


new!!! Solutions to the final exam have been posted to Section IV of this website.

The graded homework set 5 and the final exam are now available. Stop by my office any time during the spring quarter to retrieve your work.

Final grades have been posted. The final exam scores ranged from a low of 52 to a high of 91. The mean was 67.

One final bonus to celebrate the spring break. I have posted a class handout entitled Thomas Precession and the BMT equation to Section V of this website. A link to one of the references to the handout is also provided in Section IX of this website. Enjoy!

Table of Contents

[ I. General Information and Syllabus | II. Disability Statement to Students in Class | III. Problem Sets and Exams | IV. Solutions to Problem Sets and Exams | V. Other Class Handouts | VI. References on special functions of mathematical physics | VII. Free Textbooks on Classical Electrodynamics | VIII. Other books of interest | IX. Articles of interest | X. Web pages of interest ]

I. General Information and Syllabus

The General Information and Syllabus handout is available in either PDF or Postscript format     [PDF | Postscript]
Some of the information in this handout is reproduced here.

General Information

Instructor Howard Haber
Office ISB 326
Phone 459-4228
Office Hours Mondays 2--4 pm
e-mail haber@scipp.ucsc.edu
academic webpage http://scipp.ucsc.edu/~haber/

Class Hours

Lectures: Tuesdays and Thursdays, 5:10--6:45 pm, ISB 231

Required Textbook

Classical Electrodynamics, 3rd Edition, by John David Jackson     [Errata for the 18th reprinting (dated 2008) can be found here in PDF format.]

Course Grading and Requirements

50% Homework (5 problem sets)
20% Midterm Exam (24 hour take-home exam; due Friday February 24, 2017 at 7 pm)
30% Final Exam (Wednesday March 22, 2017, from 4--7 pm)

Homework assignments are not optional. Homework assignments are typically due on Thursdays (with two weeks allotted for each homework set). You are encouraged to discuss the class material and homework problems with your classmates and to work in groups, but all submitted problems should represent your own work and understanding.

The final exam will be held in ISB 231. This exam will be three hours long and cover the complete course material. You must take the final exam to pass the course. You will be permitted to consult any textbook of your choosing, your own handwritten notes, and any class handout (including solutions to the problem sets) during the final exam.

Course Syllabus

  1. Review of Maxwell's Equations
  2. Plane Electromagnetic Waves
  3. Wave Propagation in a Dispersive Medium
  4. Special Theory of Relativity
  5. Simple Radiating Systems and Antennae
  6. Multipole Fields
  7. Dynamics of Relativistic Particles and Electromagnetic Fields
  8. Radiation by Accelerated Charges
  9. Scattering of Electromagnetic Waves

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II. Disability Statement to Students in Class

UC Santa Cruz is committed to creating an academic environment that supports its diverse student body. If you are a student with a disability who requires accommodations to achieve equal access in this course, please submit your Accommodation Authorization Letter from the Disability Resource Center (DRC) to me privately during my office hours or by appointment, preferably within the first two weeks of the quarter. At this time, we would also like us to discuss ways we can ensure your full participation in the course. We encourage all students who may benefit from learning more about DRC services to contact DRC by phone at 831-459-2089 or by email at drc@ucsc.edu.

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III. Problem Sets and Exams

Problem sets and exams are available in either PDF or Postscript formats.

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IV. Solutions to Problem Sets and Exams

The problem set and exam solutions are available in either PDF or postscript formats.

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V. Other Class Handouts

Class handouts are available in either PDF or postscript formats.

1. A handout entitled The Poisson equation and the inverse Laplacian introduces the concept of the inverse Laplacian and uses it to provide a solution to Poisson's equation. This method of solution is closely related to the Green function technique to Poisson's equation. Finally, the inverse Laplacian is used to derive the Helmholtz decomposition of a vector field.   [PDF | Postscript].

2. A handout entitled The Riemann-Lebesgue Lemma is a very important result of Fourier analysis. It has many applications in mathematical physics. A simple proof of the Riemann-Lebesgue lemma is given in this handout.   [PDF | Postscript].

3. The identity 1/(x ± iε)=P(1/x)∓iπ is called the Sokhotski-Plemelj formula. In a handout entitled The Sokhotski-Plemelj Formula, three different derivations of this formula are provided. In an appendix, the Fourier transform of a tempered distribution is discussed and then applied to the Sokhotski-Plemelj formula. Since the Fourier transform of a tempered distribution (and its inverse Fourier transform) are unique, this analysis proides a fourth derivation of the Sokhotski-Plemelj formula.   [PDF | Postscript].

4. A handout entitled Examples of four-vectors examines the properties of the velocity, momentum, force and acceleration four-vectors of special relativity. These notes provide a careful derivation of the law of addition of velocities. In addition, it presents the relativistic version of Newton's 2nd Law and discusses the meaning of constant acceleration for relativistic motion.   [PDF | Postscript].

5. A handout entitled The electromagnetic fields of a uniformly moving charge provides a derivation of the expressions for the electric and magnetic fields of a uniformly moving charge in the laboratory frame. The derivation, which is based on performing a Lorentz boost from the rest frame of the charge to the laboratory frame, is more general than the one given in Jackson. Also given are alternative expressions for the electromagnetic fields in terms of quantities that depend on the retarded time.   [PDF | Postscript].

6. The handout entitled The tensor spherical harmonics treats the vector spherical harmonics in a more general context, which provides for a better understanding of their origin (and allows for further generalizations). The formalism of addition of angular momentum in quantum mechanics plays a key role in this more general treatment.   [PDF | Postscript].

7. These notes derive the classical Hamiltonian of a non-relativistic charged particle in an electromagnetic field, and introduces the principle of minimal substitution. This principle is then employed to obtain the Schrodinger equation in the presence of an external electromagnetic field. As an example, the special case of a uniform magnetic field is exhibited. Finally, we demonstrate the origin of the coupling of the spin operator to the external magnetic field in the case of a charged spin-1/2 particle.     [PDF | Postscript]

8. These notes present a derivation of the Thomas precession of a particle with intrinsic spin. The derivation is then extended to the case of a charged particle with intrinsic spin in an external electromagnetic field. The dynamics of such a particle are described by the BMT equation. A detailed derivation is provided that follows the method employed in the derivation of the Thomas precession.     [PDF | Postscript]

9. In deriving the Liénard-Wiechert potentials in class, we needed to evaluate a Jacobian determinant associated with a change of variables. To evaluate this determinant, we used the result, det(δij+aibj)=1+ ab. In this handount, three derivations of this determinantal identity are provided.     [PDF | Postscript]

10. A handout entitled The Poisson Sum Formula provides a derivation of this famous formula, along with some commentary. You will use this formula in solving problem 14.13 of Jackson, which examines the radiation of an accelerating particle whose motion is periodic.     [PDF | Postscript].

11. A handout entitled The power spectrum of Cherenkov radiation provides a derivation of the Tamm-Frank formula for the differential distribution of power with respect to the frequencies of the radiation. This derivation employs the methods of Chapter 14 of Jackson. In contrast, Jackson uses a different method for obtaining the Tamm-Frank formula in Chapter 13.   [PDF | Postscript].

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VI. References on special functions of mathematical physics

1. A superb resource for both the elementary functions and the special functions of mathematical physics is the Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun, which is freely available on-line. The home page for this resource can be found here. There, you will find links to a frames interface of the book. Another scan of the book can be found here. A third independent link to the book can be found here.

2. The NIST Handbook of Mathematical Functions (published by Cambridge University Press), together with its Web counterpart, the NIST Digital Library of Mathematical Functions (DLMF), is the culmination of a project that was conceived in 1996 at the National Institute of Standards and Technology (NIST). The project had two equally important goals: to develop an authoritative replacement for the highly successful Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, published in 1964 by the National Bureau of Standards (M. Abramowitz and I. A. Stegun, editors); and to disseminate essentially the same information from a public Web site operated by NIST. The new Handbook and DLMF are the work of many hands: editors, associate editors, authors, validators, and numerous technical experts. The NIST Handbook covers the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. All of the mathematical information contained in the Handbook is also contained in the DLMF, along with additional features such as more graphics, expanded tables, and higher members of some families of formulas. A PDF copy of the handbook is provided here:   [PDF]

3. One of the classic references to special functions is a three volume set entitled Higher Transcendental Functions (edited by A. Erdelyi), which was compiled in 1953 and is based in part on notes left by Harry Bateman. This was the primary reference for a generation of physicists and applied mathematicians, which is colloquially referred to as the Bateman Manuscripts. This esteemed reference work continues to be a valuable resources for students and professionals. PDF versions of the three volumes are now available free of charge. Check out the three volumes by clicking on the relevant links here:     [Volume 1 | Volume 2 | Volume 3].

4. Another very useful reference for both the elementary functions and the special functions of mathematical physics is An Atlas of Functions (2nd edition) by Keith B. Oldham, Jan Myland and Jerome Spanier, published by Springer Science in 2009. This resource is freely available on-line to students at the University of California at this link.

5. Yet another excellent website for both the elementary functions and the special functions of mathematical physics is the Wolfram Functions site. This site was created with Mathematica and is developed and maintained by Wolfram Research with partial support from the National Science Foundation. Additional mathematical information can be found on the related Wolfram MathWorld site.

6. One of my favorite books on special functions is Special Functions and Their Applications by N.N. Lebedev (Dover Publications, Inc., Mineola, NY, 1972). It provides invaluable information on special functions, while being extremely cheap to buy (and even cheaper to peruse on Google Books).

7. In working out Jackson problems, you will encounter many difficult integrals. Although you may be tempted to use Mathematica or Maple (which sometimes is the easiest approach), I cannot overestimate the value of a good table of integrals. Professionals always choose first to consult the Table of Integrals, Series and Products, 8th edition, by I.S. Gradshteyn and Ryzhik, edited by Daniel Zwillinger and Victor Moll (Academic Press, Elsevier, Inc., Amsterdam, 2015). This resource is freely available on-line to students at the University of California at this link.

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VII. Free Textbooks on Electrodynamics

1. Electromagnetic Field Theory by Bo Thidé is the result of a long standing advanced electrodynamics internet textbook project, roughly at the same level as Jackson. A companion book with exercises (electrodynamics problems with solutions) is available for free download too.

2. Classical Electrodynamics, Part II by Robert G. Brown is a set of notes written for a graduate electrodynamics course taught at Duke University. These notes have evolved into an online book that is available here:   [PDF | Postscript]

3. Macroscopic Electrodynamics, by Walter Wilcox and Chris Thron is the online version of a book by the same name that was published in March, 2016 by World Scientific.

4. Electromagnetic Waves and Antennas, by Sophocles J. Orfanidis is freely available here:   [PDF]

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VIII. Other books of interest

1. One of the important mathematical tools used in classical electrodynamics is the theory of complex variables and analytic functions.There are many books that treat this topic. One very nice treatment can be found in Chapter 6 of Frederick W. Byron and Robert C. Fuller, Mathematics of Classical and Quantum Physics (Dover Publications, Inc., New York, 1992), originally published by the Addison-Wesley Publishing Company in 1970, but has now been reprinted in an inexpensive paperback edition by Dover Publications. Section 6.6 provides a pedagogical introduction to dispersion relations, which will be especially useful for understanding the material of Section 7.10 of Jackson. For your convenience, I am providing a link to Chapter 6 here.   [PDF]

2. A somewhat terse but useful introduction to complex integration can be found in Chapter 2 of Debabrata Basu, Introduction to Classical and Modern Analysis and Their Application to Group Representation Theory (World Scientific, Singapore, 2011). The entire chapter is provided free of charge by Google Books. The treatment of analytic continuation at the beginning of Chapter 3 is especially noteworthy. In particular, the example provided in Section 3.1 is spectacular and deserves a place in any book on complex analysis!

3. Many fully solved problems in the theory of complex variables can be found in Schaum's outlines, Complex Variables (2nd edition) by Murray R. Spiegel, Seymour Lipschutz, John J. Schiller and Dennis Spellman (McGraw Hill, New York, 2009).   [PDF]

4. Fianlly, you can also peruse Chapter 1 of Alexander O. Gogolin, Lectures on Complex Integration, which surveys the basics of complex analysis and applies it to evaluating integrals of various types. This resource is available to University of California students.   [PDF]

5. One of the best elementary treatments of Green functions for partial differential equations suitable for physicists can be found in a book by Gabriel Barton, Elements of Green's Functions and Propagation---Potentials, Diffusion and Waves, published by Oxford Science Publications in 1989. Among other things, there is a very clear discussion of which boundary conditions constitute a well-posed problem.

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IX. Articles of interest

1. Davon Ferrara, who was once a physics graduate student at Vanderbilt University claimed that everything he needed to know in life he learned from Jackson Electrodynamics. To verify his assertion, he posted the following document, available here in your choice of formats:     [PDF | [DOCX]

2. The Lorentz gauge condition, which was specified by the Dutch theoretical physicist Hendrik Antoon Lorentz in 1904 was not the first to write down this condition. Thirty-seven years earlier in 1867, the Danish theorist Ludvig Valentin Lorenz introduced a similar constraint on the choice of scalar and vector potentials. J.D. Jackson argues that Lorenz deserves the recognition of understanding the arbitrariness and equivalence of difference forms of the potential. For more details check out his paper, Examples of the zeroth theorem of the history of science, which can be found in J.D. Jackson, American Journal of Physics 76, 704 (2008).     [PDF]

Jackson is not the first to point out the slighting of Lorenz. In 1991, J. van Bladel also argued in Lorenz' favor in Lorenz or Lorentz?, which can be found in J. van Bladel, Antennas and Propagation Magazine, IEEE 33, Issue 2, 69 (1991).     [PDF]

3. The historical roots of gauge invariance are described in this scholarly review by J.D. Jackson and L.B. Okun, Review of Modern Physics 73, 663 (2001). The relevant work of L.V. Lorenz and H.A. Lorentz are also clarified in this survey.     [PDF]

4. J.D. Jackson discusses how to gauge transform from one set of scalar and vector potentials to another set in a paper entitled, From Lorenz to Coulomb and other explicit gauge transformations, which can be found in J.D. Jackson, American Journal of Physics 70, 917 (2002). This paper also shows that the electric and magnetic fields display the properties of causality and propagation at the speed of light, even if these properties are not exhibited by the corresponding scalar and vector potentials.     [PDF]

5. A comparison of the causal relations in the Lorenz and Coulomb gauges is presented in an article entitled, in Causality in the Coulomb Gauge, which can be found in O.L. Brill and B. Goodman, American Journal of Physics 35, 832 (1967).     [PDF]

6. In Jackson problem 7.27, the term identified as the spin-angular momentum of the electromagnetic field is an expression that is not invariant under gauge transformations. However, one can show that a gauge-invariant expression for the spin-angular momentum does exist that reduces in the Coulomb gauge to Jackson's expression. For further details check out Canonical separation of angular momentum of light into its orbital and spin parts, which can be found in Iwo Bialynicki-Birula and Zofia Bialynicka-Birula, Journal of Optics 13, 064014 (2011).     [PDF]

7. In class, I stated that the only invariant Lorentz tensors of special relativity are δμν, gμν, gμν, the Levi Civita tensor εμναβ (and its relatives with raised indices), and product thereof. A similar theorem exists for n-dimensional Cartesian tensors, where the only invariant tensors (also called isotropic tensors) are δij, εi1...in and products thereof. For a simple proof of the latter theorem, see On isotropic tensors by Sir Harold Jeffreys, Mathematical Proceedings of the Cambridge Philosophical Society 73, 173--176 (1973).     [PDF]
A similar proof can be given for the Lorentz tensors of special relativity as well as tensors with respect to more general transformations. A more sophisticated treatment of the classification of invariant tensors can be found in On the classification of isotropic tensors by P.G. Appleby, B.R. Duffy and R.W. Ogden, Glasgow Mathematical Journal 29, 185--196 (1987).     [PDF]

8. A pedagogical paper that treats the properties of localized steady charges and currents moving with a constant velocity provides the exact relativistic transformations for the electric and magnetic dipole moments (which were treated to first order in β in Jackson, problem 11.27). Further details can be found in the solutions to problem set 2. But, you might be interested in having a look at the original paper entitled The Electric Dipole Moment of a Moving Magnetic Dipole by George P. Fisher, American Journal of Physics 39, 1528--1533 (1971).     [PDF].

A related paper of interest that treats the same subject matter in more depth is entitled Moving pointlike charges and electric and magnetic dipoles by Marijan Ribarič and Luka Šušteršič, American Journal of Physics 60, 513--519 (1992).     [PDF]

Remarkably, the result obtained in part (a) of Jackson, problem 11.27 often appears in the literature without the overall factor of 1/2. For a discussion on this discrepancy, see the paper entitled Magnetic dipole moment of a moving electric dipole by V. Hnizdo, American Journal of Physics 80, 645--647 (2012).     [PDF]
Further analysis is provided by V. Hnizdo and Kirk T. MacDonald, Fields and moments of a moving electric dipole     [PDF | Postscript]

9. A divergenceless vector field F(r) can be represented in terms of two scalar potential fields ψ(r) and χ(r) [called Debye potentials] as F(r)= Lψ(r)+∇×Lχ(r), where L = -i r×∇. A proof of this result aimed at physicists can be found in a paper entitled Debye potential representation of vector fields by C.G. Gray and B.G. Nickel, American Journal of Physics 46, 735--736 (1978).     [PDF]

The application of the Debye potentials to the multipole expansion can be found in a paper entitled Multipole expansions of electromagnetic fields using Debye potentials by C.G. Gray, American Journal of Physics 46, 169--179 (1978). In particular, Appendix A of this paper provides a proof of the expansion of an arbitrary vector field (not assumed to be divergenceless) in terms of three Debye potentials, and Appendix F of this paper contains a number of very useful identities involving the differential operators and L.     [PDF]

10. The role of the observer is frequently obscured, either by writing equations in a coordinate system implicitly pertaining to some specific observer or by entangling the invariance and the observer dependence of physical quantities. For example, this confusion often arises in the treatment of Thomas precession. The confusion underlying the aforementioned misconceptions are clarified by a number of examples in relativistic kinematics and classical electrodynamics in the paper, Bruno Klajn and Ivica Smolić, entitled Subtleties of Invariance, Covariance and Observer Independence, European Journal of Physics 34, 887--899 (2013). In particular, check out the very nice analysis of the Thomas precession which is presented there.     [PDF]

11. A simple and pedagogical derivation of the Bargmann-Michel-Telegdi (BMT) equation (which also exhibit the presence of Thomas precession) is given in a paper by Krysztof Rębilas, entitled Simple approach to relativistic spin dynamics, American Journal of Physics 79, 1064--1067 (2011).     [PDF]

12. Using Jefimenko's equations, one can derive directly the expressions for the electric and magnetic fields of a moving point charge without having to first calculate the scalar and vector potentials. This calculation is carried out in detail in Omer Dushek and Sergiy V. Kuzmon, The fields of a moving point charge: a new derivation from Jefimenko's euations, European Journal of Physics 25, 343--349 (2004).     [PDF]

13. In class, we derived the Liénard-Wiechert scalar and vector potentials for an accelerating point charge, and then evaluated the corresponding electric and magnetic fields. Our analysis was performed under the assumption that the mass of the charge is non-zero (so that v<c). The electric and magnetic fields due to a massless accelerating point charge have been obtained for the first time in Francesco Azzurli and Jurt Lechner, The Liénard-Wichert field of accelerated massless charges, Physics Letters A 377, 1025--1029 (2013).     [PDF]

14. A point charge moving with constant velocity in a material medium generates time-dependent elemental dipoles. If the velocity of the charge is larger than c/nr (i.e. the speed of light in the medium), then there is a coherent superposition of the radiation emitted by each elemental dipole. The total coherent radiation can be identified as the Cherenkov radiation. This analysis provides for a very physical picture for the origin of Cherenkov radiation. For further details, see the article entitled Induced time-dependent polarization and the Cherenkov effect by J.A.E. Roa-Neri, J.L. Jimenez and M. Villavicencio, European Journal of Physics, 16, 191--194 (1995).     [PDF]

15. In the computation of Cherenkov radiation presented in class, the electric field is singular on the surface of the Mach cone where nv=ct/n and r=ct/n, and n is the index of refraction of the medium, These singularities arise due to an idealization of the problem (e.g. the assumption of a point charge); in a more realistic setting these singularities are smoothed out. For example, see the paper entitled Cherenkov radiation from a charge of finite size or a bunch of charges, by Glenn S. Smith, American Journal of Physics 61, 147--155 (1993).     [PDF]

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X. Web pages of interest

1. One of the most prolific sources for pedagogical treatments of a variety of interesting problems in classical electrodynamics is a webpage maintained by Kirk T. McDonald entitled Physics Examples and other Pedagogic Diversions. Although some of the content is password protected, most of the articles listed are available freely.

2. Which direction is the electric field rotating in a left or right circularly polarized wave? Check out the Wikipedia page on circular polarization, which provides some enlightening animations that illustate the answer to this question.

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Last Updated: March 27, 2017