Electromagnetism in 2+1 spacetime dimensions was the subject of the first problem on the final exam. If you would like further information on this very interesting topic, check out some of the papers that have been posted to Section X of this website.
Please stop by my office any time during the spring quarter to retrieve your work. Enjoy your spring break!
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 3--4 pm and Thursdays 2--3 pm, or by appointment | |
haber@scipp.ucsc.edu | ||
academic webpage | http://scipp.ucsc.edu/~haber/ | |
Lectures: Tuesdays and Thursdays, 11:40 am -- 1:15 pm
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 (take-home exam--pick up in class on Tuesday February 20 and return to class on Thursday February 22)
30% Final Exam (Wednesday March 20, 2024, from 4--7 pm)
Homework assignments are not optional. Homework assignments are typically due on Thursdays (with two weeks allotted for each homework set) unless otherwise noted. 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.
Problem sets and exams are available in either PDF or postscript formats.
The problem set and exam solutions 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 Generalized Functions for Physics provides a practical introduction to generalized functions (also called distributions) that are particularly useful in many mathematical physics applications. We focus primarily on identifying the most common generalized functions employed in physics and methods for manipulating them. Proofs of the Riemann-Lebesgue Lemma and the Poisson sum formula are also provided. [PDF | Postscript].
3. A handout entitled Polarization Vectors and Polarization Sums examines the properties of the polarization vectors that are used to characterize the polarization of an electromagnetic wave. An explicit form for the polarization vector of a left-circularly and right-circularly polarized wave is given in the case where the wave propagates in an arbitrary direction. In electromagnetic scattering processes, if the polarization of the incoming and outgoing waves are not observed, then one must average over the initial polarization states and sum over the final polarization states. A formula for performing polarization sums is derived in this note and is applied to Thomson scattering as an example. [PDF | Postscript].
4. A handout entitled Evaluating f(τ) using complex integration provides a detailed derivation of the function f(τ) that appears in the relation between the time-dependent electric displacement field and the electric field in a linear, homogeneous and isotropic medium. [PDF | Postscript].
5. A handout entitled Proof that an electromagnetic wave does not propagate faster than c in a dispersive medium demonstrates that even though it is possible in a dispersive medium for the phase velocity and/or the group velocity to exceed c, one can show that the wave front of an electromagnetic wave cannot propagate faster than c. [PDF | Postscript].
6. 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 definition of constant acceleration for relativistic motion. [PDF | Postscript].
7. 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].
8. Consider a unit vector n pointing in the radial direction, with components, n_{i} (for i=1,2,3). How do we evaluate the integral of the product n_{i} n_{j} · · · n_{k} over solid angles? The handout entitled, Evaluation of some integrals over solid angles, provides an elegant evaluation for the case where two or four factors of n appear in the integrand. Moreover, it is shown that for an odd number of factors of n, the resulting integral over solid angles vanishes. [PDF | Postscript].
9. In analyzing electric dipole radiation from a charge that is rotating in the x-y plane, one can arrive at results that seem to contradict each other if one is not careful. Some of these issues are explored in problem 9.1 of Jackson. I provide here a detailed solution to this problem to demonstrate how the apparent contradictions are resolved. [PDF | Postscript]
10. It is convenient to expand the Green function of the three-dimensional Helmholtz equation in terms of the spherical harmonics. The radial coefficient of this expansion is called the radial Green function, and it satisfies the differential equation of the spherical Bessel functions after replacing the zero on the right hand side with a delta function source. A detailed derivations of the Green function of the Helmholtz equation and the corresponding radial Green function are provided in this class handout entitled The Radial Green Function. [PDF | Postscript].
11. 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. The application of vector spherical harmonics to the electromagnetic multipole radiation fields is provided. [PDF | Postscript].
12. The Lagrangian in classical mechanics is a function of the generalized coordinates and the time derivatives of the coordinates. In contrast, the Lagrangian density that arises in field theory is a function of the fields, the time derivatives of the fields and the space derivatives of the fields. The appearance of the space derivatives of the fields is a consequence of local interactions. In particular, this holds true both in nonrelativistic and relativistic field theory. To better understand this point it is instructive to derive a model one dimensional field theory as a limit of a system of n interacting bodies as n → ∞. This is beautifully explained in a book by Bjørn Felsager entitled Geometry, Particles, and Fields (Springer-Verlag, New York, 1998). Six relevant pages from this book are provided here: [PDF]
13. 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]
14. This handout 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. Finally, we discuss the famous factor of two discrepancy in the standard "derivation" in quantum mechanics textbooks of the spin-orbit Hamiltonian that governs the hydrogen atom. [PDF | Postscript]
15. 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]
16. The handout entitled, Evaluation of some integrals over solid angles--Part 2, provides a derivation of three angular integrals that are used to compute the total power radiated by a relativistic accelerating charge. After providing an explicit derivation of the integrals, a second method is suggested that makes use of the covariance properties of Euclidean tensors, which was employed in the previous handout on the evaluation of some integrals over solid angles. [PDF | Postscript].
17. 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, which avoids the appearance of δ(0) that was obtained in our class derivation. Note that 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.
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, [Volumes I--III] (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 | Errata--Volume 1 | Errata--Volume 2]
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. [PDF]
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. A second edition of this book is scheduled to be published in February, 2024
4. Electromagnetic Waves and Antennas, by Sophocles J. Orfanidis is freely available here: [PDF]
5. Pawel Klimas has provided on his webpage access to his lecture notes on classical electrodynamics. For your convenience, I have collected these notes in a single PDF file, which is provided here: [PDF]
6. Nicholas Wheeler, who taught physics at Reed College for many years left a valuable collection of his lecture notes which can be accessed from his webpage. Among the many documents available to the public, I found an almost completed textbook entitled Principles of Classical Electrodynamics--A Laptop Text.
1. Chapter 19 of the book by Frederick W. King, Hilbert Transforms, Volume 2 provides a very nice introduction to the Kramers-Kronig relations and their applications to dispersive media in the study of electromagnetism. This resource is available to University of California students. [PDF]
2. Perhaps you heard in some previous class that you cannot treat accelerated reference frames in special relativity without extending the analysis to general relativity. If so, you have been seriously misinformed. Accelerated reference frames can be treated consistently within the confines of special relativity. An excellent book that shows you how to do this is by Eric Gourgoulhon, Special Relativity in General Frames--From Particles to Astrophysics (Springer-Verlag, Berlin, Germany, 2013). This resource is available to University of California students. [PDF]
3. 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 nice discussion of the Helmholtz decomposition and the Coulomb gauge is given by Kirk T. McDonald, The Helmholtz Decomposition and the Coulomb Gauge. [PDF]
6. The uniqueness of the Helmholtz decomposition relies on a requirement that the vector field vanish sufficiently fast at infinity. Two relevant papers that discuss the meaning of "vanish sufficiently fast'' are listed below.
7. 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]
8. 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 products 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]
9. The motion of an observer with constant proper acceleration is called relativistic hyperbolic motion. Such motion exhibits a nonzero third and fourth order proper time derivative of the motion, known as jerk and snap, respectively. For further details, check out the following two papers:
10. In problem 1 on problem set 2, you computed exp(- ζ β·K) and showed that it is equal to the boost matrix defined in eq. (11.98) of Jackson. It would be interesting to obtain a closed-form expression for the most general proper orthochronous Lorentz transformation by evaluating the exponential exp(- θ n·S - ζ β·K). You can find the result of this calculation in a paper that I recently posted to the arXiv entitled Explicit form for the most general Lorentz transformation revisited, arXiv:2312.12969 [physics.class-ph] [PDF]
11. 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]
12. Unlike electromagnetic radiation, the multipole expansion of gravitational radiation begins with the quadrupole. Nevertheless, one can learn much about gravitational radiation by comparing with electromagnetic quadrupole radiation. A recent pedagogical introduction to some key computations in gravitational waves via a side-by-side comparison with the quadrupole contribution of electromagnetic radiation is presented in a paper entitled An introduction to gravitational waves through electrodynamics: a quadrupole comparison by Glauber C. Dorsch and Lucas E.A. Porto, European Journal of Physics 43, 025602 (2022). [PDF]
13. In Chapter 9 of Jackson, formulae for the power radiated in electric dipole, magnetic dipole and electric quadrupole radiation are derived under the assumption that the sources are harmonic. In class, I derived the more general expressions for the power in cases where the sources have an arbitrary dependence on time. My derivations were based on the Jefimenko equations for the electric and magnetic fields. One can also employ Jackson's approach by starting from an expression for the vector potential. Details of this latter technique can be found in a paper entitled Radiation of the electromagnetic field beyond the dipole approximation by Andrij Rovenchak and Yuri Krynytskyi, American Journal of Physics 86, 727--732 (2018). [PDF]
14. 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]
15. 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 that is presented there. [PDF]
16. 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]
17. 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]
18. 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]
For further details, you can check out
Chapter 17 of Kurt Lechner, Classical
Electrodynamics--A Modern Perspective (Springer International
Publishing, Cham, Switzerland, 2018).
[PDF]
19. 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 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]
20. In the computation of Cherenkov radiation presented in class, the electric field is singular on the surface of the Mach cone where n•v=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]
21. In class, we saw that the electric and magnetic fields were components of a rank-2 antisymmetric tensor in 3+1 dimensional spacetime. It is interesting to see what happens if you formulate electrodynamics in 2+1 or 1+1 dimensional spacetime. Electromagnetism in 2+1 spacetime dimensions was the subject of the first problem on the final exam. If you would like further information on this very interesting topic, check out some of the papers listed below.
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 illustrate the answer to this question.
3. Hitler learns Jackson E&M. One of the strangest YouTube Videos you may encounter, with some relevance to Physics 214. [Video]
4. Quora attempts to provide an answer to the following question: Theoretical physicists, what is the most difficult class you had to take? Click here for the responses.