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!

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 | |

haber@scipp.ucsc.edu | ||

academic webpage | http://scipp.ucsc.edu/~haber/ | |

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

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

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.

- Review of Maxwell's Equations
- Plane Electromagnetic Waves
- Wave Propagation in a Dispersive Medium
- Special Theory of Relativity
- Simple Radiating Systems and Antennae
- Multipole Fields
- Dynamics of Relativistic Particles and Electromagnetic Fields
- Radiation by Accelerated Charges
- Scattering of Electromagnetic Waves

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

- Problem Set #1--due: Thursday, January 26, 2017 [PDF | Postscript]; Link to Jackson problems appearing in Problem Set #1 [PDF]
- Problem Set #2--due: Thursday, February 9, 2017 [PDF | Postscript]; Link to Jackson problems appearing in Problem Set #2 [PDF]
- Problem Set #3--due: Thursday, February 23, 2017 [PDF | Postscript]; Link to Jackson problems appearing in Problem Set #3 [PDF]
- Midterm Exam--due: Monday, February 27, 2017 [PDF | Postscript]
- Problem Set #4--due: Thursday, March 9, 2017 [PDF | Postscript]; Link to Jackson problems appearing in Problem Set #4 [PDF]
- Problem Set #5--due: Tuesday, March 21, 2017 [PDF | Postscript]; Link to Jackson problems appearing in Problem Set #5 [PDF]
- Final Exam--Wednesday, March 22, 2017 [PDF | Postscript]

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

- Solution Set #1 [PDF | Postscript]
- Solution Set #2 [PDF | Postscript]
- Solution Set #3 [PDF | Postscript]
- Midterm Exam Solutions [PDF | Postscript]
- Solution Set #4 [PDF | Postscript]
- Solution Set #5 [PDF | Postscript]
- Final Exam Solutions [PDF | Postscript]

1. A handout entitled

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(δ_{ij}+a_{i}b_{j})=1+**
a•b**. 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].

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.

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]

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.

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 δ_{μ}^{ν},
*g*_{μν}, *g*^{μν},
the Levi Civita tensor ε_{μναβ}
(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
δ_{ij}, ε_{i1}..._{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**)=
**L**ψ(**r**)+**∇×L**χ(**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/n _{r}* (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

15. In the computation of Cherenkov radiation presented in class, the
electric field is singular on the surface of the Mach cone where
** n•v**=

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.

haber@scipp.ucsc.edu

Last Updated: March 27, 2017