This page contains copies of the class handouts, and other pertinent items ofinterest for the U.C. Santa Cruz Physics 214 graduate Electromagnetism II class.
- Classical Electrodynamics Jackson 3rd Solutions
- Classical Electrodynamics Jackson 2nd Edition Solutions Manual 2017
- Classical Electrodynamics 3rd Edition Pdf
- Classical Electrodynamics Jackson 2nd Edition Solutions Manual Download
- Jackson Electrodynamics Solution Manual
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Solution Manual A First Course in Probability Theory, 6th edition, by S. Solution Manual A First Course in String Theory, 2004, Barton Zwiebach Solution Manual A First Course in the Finite Element Method, 4th Edition logan Solution Manual A Practical Introduction to Data Structures and Algorithm Analysis 2Ed by Shaffer Solution Manual A. Classical Electrodynamics 3rd ed - J.D. Jackson Solutions Manual.pdf.
Solutions to the final exam have been posted toSection IV of this website.The graded homework set 5 and the final exam are now available. Stopby my office any time during the spring quarter to retrieve your work.
Final grades have been posted. The final exam scores ranged from alow of 52 to a high of 91. The mean was 67.
One final bonus to celebrate the spring break. I have posted a classhandout entitled Thomas Precession and the BMT equation toSection V of this website. A link to one of the referencesto the handout is also provided in Section IX ofthis 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 | |
| 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 inPDF 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 assignmentsare 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 coursematerial. You must take the final exam to pass the course.You will be permitted to consult any textbook of your choosing, your ownhandwritten notes, and any class handout (including solutions to theproblem sets) during the final exam.
Course Syllabus
- 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
II. Disability Statement to Students in Class
UC Santa Cruz is committed to creating an academic environment thatsupports its diverse student body. If you are a student with adisability who requires accommodations to achieve equal access in thiscourse, please submit your Accommodation Authorization Letter from theDisability Resource Center (DRC) to me privately during my officehours or by appointment, preferably within the first two weeks of thequarter. At this time, we would also like us to discuss ways we canensure your full participation in the course. We encourage allstudents who may benefit from learning more about DRC services tocontact DRC by phone at 831-459-2089 or by email atdrc@ucsc.edu.III. Problem Sets and Exams
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]
IV. Solutions to Problem Sets and Exams
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]
V. Other Class Handouts
Class handouts are available in either PDF or postscript formats.Classical Electrodynamics Jackson 3rd Solutions
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 techniqueto Poisson's equation. Finally, the inverse Laplacian isused to derive the Helmholtz decomposition of a vector field. [PDF| Postscript].
Classical Electrodynamics Jackson 2nd Edition Solutions Manual 2017
2. A handout entitled TheRiemann-Lebesgue Lemma is a very important result ofFourier analysis. It has many applications in mathematical physics. A simple proof of the Riemann-Lebesgue lemmais given in this handout. [PDF| Postscript].
3. The identity 1/(x ± iε)=P(1/x)∓iπ is calledthe Sokhotski-Plemelj formula. In a handout entitled TheSokhotski-Plemelj Formula, three different derivations of thisformula are provided. In an appendix, the Fourier transform of atempered distribution is discussed and then applied to theSokhotski-Plemelj formula. Since the Fourier transform of a tempereddistribution (and its inverse Fourier transform) are unique, thisanalysis proides a fourth derivation of the Sokhotski-Plemelj formula. [PDF| Postscript].
4. A handout entitled Examples offour-vectors examines the properties of thevelocity, momentum, force and acceleration four-vectors of specialrelativity. These notes provide a careful derivation of the law of addition of velocities. Inaddition, it presents the relativistic version of Newton's 2nd Lawand 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 ofthe expressions for the electric and magnetic fields of a uniformlymoving charge in the laboratory frame. The derivation, which is basedon performing a Lorentz boost from the rest frame of the charge tothe laboratory frame, is more generalthan the one given in Jackson. Also given are alternative expressions for theelectromagnetic fields in terms of quantities that depend on theretarded time. [PDF| Postscript].
6. The handout entitled The tensor spherical harmonics treats thevector spherical harmonics in a more general context,which provides for a better understanding of their origin (and allows forfurther generalizations). The formalism of addition of angular momentum in quantum mechanics plays a key role in this more generaltreatment. [PDF| Postscript].
7. These notes derive the classical Hamiltonian of a non-relativisticcharged particle in an electromagnetic field, and introduces theprinciple of minimal substitution. This principle is then employed to obtain the Schrodinger equation inthe 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 fieldin the case of a charged spin-1/2 particle. [PDF | Postscript]
8. These notes present a derivation of the Thomas precession of aparticle with intrinsic spin. The derivation is then extended to thecase of a charged particle with intrinsic spin in an externalelectromagnetic field. The dynamics of such a particle are describedby the BMT equation. A detailed derivation is provided that followsthe method employed in the derivation of the Thomas precession. [PDF | Postscript]
9. In deriving the Liénard-Wiechert potentials in class, we needed toevaluate a Jacobian determinant associated with a change of variables.To evaluate this determinant, we used the result,det(δij+aibj)=1+a•b. In this handount, three derivations of this determinantalidentity are provided. [PDF | Postscript]
10. A handout entitled The Poisson Sum Formula provides a derivation of this famous formula, along with somecommentary. You will use this formula in solving problem 14.13 ofJackson, which examines the radiation of an accelerating particlewhose motion is periodic. [PDF| Postscript].
11. A handout entitled The power spectrum ofCherenkov radiation provides a derivation of the Tamm-Frankformula for the differential distribution of power with respect tothe frequencies of the radiation. This derivation employs the methodsof Chapter 14 of Jackson. In contrast, Jackson uses a differentmethod for obtaining the Tamm-Frank formula in Chapter 13. [PDF| Postscript].
VI. References on special functions of mathematical physics
1. A superb resource for both the elementary functions and thespecial functions of mathematical physics is theHandbookof Mathematical Functions by Milton Abramowitz and Irene A. Stegun,which is freely available on-line. The home page for thisresource can be foundhere. There, youwill find links to a framesinterface of the book. Another scan of the book can be found here.A third independent link to the book can be foundhere.
2. The NISTHandbook of Mathematical Functions (publishedby Cambridge University Press), together with its Webcounterpart, the NIST Digital Library of Mathematical Functions(DLMF), is the culmination of a project that was conceived in 1996 atthe National Institute of Standards and Technology (NIST). The projecthad two equally important goals: to develop an authoritativereplacement for the highly successful Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables, published in1964 by the National Bureau of Standards (M. Abramowitz andI. A. Stegun, editors); and to disseminate essentially the sameinformation from a public Web site operated by NIST. The new Handbookand DLMF are the work of many hands: editors, associate editors,authors, validators, and numerous technical experts.The NISTHandbook coversthe properties of mathematical functions, from elementarytrigonometric functions to the multitude of special functions.All of the mathematical information contained in the Handbook is alsocontained in the DLMF, along with additional features such as moregraphics, expanded tables, and higher members of some families offormulas. A PDF copy of the handbook is provided here: [PDF]
3. One of the classic references to special functions is a threevolume set entitled Higher Transcendental Functions (edited by A. Erdelyi), which wascompiled in 1953 and is based in part on notes left by Harry Bateman.This was the primary reference for a generation of physicists andapplied mathematicians, which is colloquially referred to as theBateman 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 thespecial functions of mathematical physics is An Atlasof Functions (2nd edition) by Keith B. Oldham, Jan Myland and JeromeSpanier, published by Springer Science in 2009.This resource is freely available on-line to studentsat the University of California at thislink.
5. Yet another excellent website for both the elementary functions and thespecial functions of mathematical physics is the Wolfram Functionssite. This site was created with Mathematica and is developed and maintainedby Wolfram Research with partial support from the National Science Foundation. Additional mathematical information canbe found on the relatedWolfram MathWorld site.
6. One of my favorite books on special functions isSpecial Functions and Their Applications by N.N. Lebedev (Dover Publications, Inc., Mineola, NY, 1972). Itprovides invaluable information on special functions, while being extremely cheap to buy (and even cheaper to peruse onGoogle Books).
7. In working out Jackson problems, you will encounter many difficultintegrals. Although you may be tempted to use Mathematica or Maple(which sometimes is the easiest approach), I cannot overestimate thevalue of a good table of integrals. Professionals always choose firstto consult the Table ofIntegrals, Series and Products, 8th edition,by I.S. Gradshteyn and Ryzhik, edited byDaniel Zwillinger and Victor Moll (Academic Press, Elsevier, Inc.,Amsterdam, 2015). This resource is freely available on-line to studentsat the University of California at thislink.
VII. Free Textbooks on Electrodynamics
1. Electromagnetic Field Theoryby Bo Thidé is the result of a long standing advancedelectrodynamics internet textbook project, roughly at the same levelas Jackson. A companion book with exercises (electrodynamics problems with solutions) is available for free download too.
2. ClassicalElectrodynamics, 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. MacroscopicElectrodynamics, by Walter Wilcox and Chris Thron is the onlineversion of a book by the same name that was published in March, 2016by World Scientific.
4. ElectromagneticWaves and Antennas, by Sophocles J. Orfanidis is freelyavailable here: [PDF]
VIII. Other books of interest
1. One of the important mathematical tools used in classical electrodynamicsis the theory of complex variables and analytic functions.There aremany books that treat this topic. One very nice treatment can befound 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-WesleyPublishing Company in 1970, but has now been reprinted in aninexpensive paperback edition by Dover Publications.Section 6.6 provides a pedagogical introduction to dispersion relations, whichwill be especially useful for understanding the material of Section 7.10of Jackson. For yourconvenience, I am providing a link to Chapter 6 here. [PDF]
2. A somewhat terse but useful introduction to complex integration canbe found in Chapter 2 of Debabrata Basu, Introductionto Classical and Modern Analysis and Their Application to GroupRepresentation Theory (World Scientific, Singapore, 2011). Theentire chapter is provided free of charge byGoogleBooks. The treatment of analytic continuation at the beginning ofChapter 3 is especially noteworthy. In particular, the exampleprovided in Section 3.1 is spectacular and deserves a place in anybook on complex analysis!
3. Many fully solved problems in the theory of complex variables canbe found in Schaum's outlines, Complex Variables (2nd edition)by Murray R. Spiegel, Seymour Lipschutz, John J. Schiller and DennisSpellman (McGraw Hill, New York, 2009). [PDF]
4. Fianlly, you can also peruse Chapter 1 of Alexander O. Gogolin, Lectureson Complex Integration, which surveys the basics of complexanalysis and applies it to evaluating integrals of various types. This resource isavailable to University of California students. [PDF]
5. One of the best elementary treatments of Green functions forpartial 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 byOxford Science Publications in 1989.Among other things, there is a very clear discussion of which boundary conditions constitute a well-posed problem.
IX. Articles of interest
1. Davon Ferrara, who was once a physics graduate student at VanderbiltUniversity claimed thateverything he needed to knowin life he learned from Jackson Electrodynamics. To verify hisassertion, he posted the following document, available here inyour 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, Examplesof the zeroth theorem of the history of science, which canbe 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. In1991, J. van Bladel also argued in Lorenz' favor inLorenz or Lorentz?, which can be found inJ. van Bladel, Antennas and Propagation Magazine, IEEE 33, Issue 2, 69 (1991). [PDF]
3. The historicalroots of gauge invariance are described in this scholarly reviewby J.D. Jackson and L.B. Okun, Review of Modern Physics 73,663 (2001). The relevant work of L.V. Lorenz and H.A. Lorentzare 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 displaythe 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 Coulombgauges is presented in an article entitled,in Causality in the Coulomb Gauge, which can be found inO.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 ofthe electromagnetic field is an expression that is not invariant undergauge transformations. However, one can show that a gauge-invariantexpression for the spin-angular momentum does exist that reduces in the Coulomb gauge to Jackson's expression. For further detailscheck 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 specialrelativity are δμν, gμν, gμν,the Levi Civita tensor εμναβ(and its relatives with raised indices), and product thereof.A similar theorem exists for n-dimensionalCartesian tensors, where the onlyinvariant tensors (also called isotropic tensors) are δij, εi1...in andproducts thereof. For a simple proof of the latter theorem, seeOn isotropic tensors by Sir Harold Jeffreys, MathematicalProceedings of the Cambridge Philosophical Society 73, 173--176 (1973). [PDF]
A similar proof can be given for the Lorentz tensors of special relativityas well as tensors with respect to more general transformations. A more sophisticatedtreatment of the classification of invariant tensors can be found inOn the classification of isotropic tensors byP.G. Appleby, B.R. Duffy and R.W. Ogden, Glasgow Mathematical Journal 29, 185--196 (1987). [PDF]
8. A pedagogicalpaper that treats the properties of localized steady charges and currents moving with a constant velocity provides the exactrelativistic transformations for the electric and magnetic dipole moments(which were treated to first order in β in Jackson, problem11.27). Further details can be found in the solutions to problem set2. But, you might be interested in having a look at the original paperentitled The Electric Dipole Moment of a Moving Magnetic Dipole byGeorge P. Fisher, American Journal of Physics 39, 1528--1533 (1971). [PDF].
A related paper of interest that treats the same subject matter inmore depth is entitled Moving pointlike charges and electric and magnetic dipoles byMarijan Ribarič and Luka Šušteršič, American Journal of Physics 60, 513--519 (1992). [PDF]
Remarkably, the result obtained in part (a) of Jackson, problem11.27 often appears in the literature without the overall factor of1/2. For a discussion on this discrepancy, see the paperentitled 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) andχ(r) [called Debye potentials] as F(r)= Lψ(r)+∇×Lχ(r), where L = -i r×∇. A proof of this result aimed atphysicists can be found in a paper entitled Debye potential representation of vector fields by C.G. Gray andB.G. Nickel, American Journal of Physics 46, 735--736 (1978). [PDF]
The application of the Debye potentials to the multipole expansion canbe found in a paper entitled Multipole expansions of electromagnetic fields using Debye potentials byC.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 tobe divergenceless) in terms of three Debye potentials, and Appendix F of this paper contains a number of veryuseful identities involving the differential operators ∇and L. [PDF]
10. The role of the observer is frequently obscured, either by writingequations in a coordinate system implicitly pertaining to somespecific observer or by entangling the invariance and the observerdependence of physical quantities. For example, this confusion often arises in the treatment of Thomas precession. The confusion underlying the aforementioned misconceptions areclarified by a number of examples in relativistic kinematics andclassical electrodynamics in thepaper, Bruno Klajn and Ivica Smolić,entitled Subtleties of Invariance, Covariance and Observer Independence, EuropeanJournal of Physics 34, 887--899 (2013). In particular, check out the very nice analysis ofthe 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 Simpleapproach to relativistic spin dynamics, American Journal ofPhysics 79, 1064--1067 (2011). [PDF]
12. Using Jefimenko's equations, one can derive directly theexpressions for the electric and magnetic fields of a moving pointcharge without having to first calculate the scalar and vectorpotentials. This calculation is carried out in detail in Omer Dushekand Sergiy V. Kuzmon, The fields of a moving point charge: a newderivation 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 thecorresponding electric and magnetic fields. Our analysis wasperformed under the assumption that the mass of the charge is non-zero(so that v<c). The electric and magnetic fields due to amassless accelerating point charge have been obtained for thefirst time in Francesco Azzurli and Jurt Lechner,The Liénard-Wichert field of accelerated massless charges, Physics Letters A377, 1025--1029 (2013). [PDF]
14. A point charge moving with constant velocity in a material mediumgenerates time-dependent elemental dipoles. If the velocityof the charge is larger than c/nr (i.e. the speed oflight in the medium), then there is a coherent superposition of theradiation emitted by each elemental dipole. The total coherentradiation can be identified as the Cherenkov radiation. This analysisprovides for a very physical picture for the origin of Cherenkovradiation. For further details, see the article entitledInduced time-dependent polarization and the Cherenkov effectby J.A.E. Roa-Neri, J.L. Jimenez and M. Villavicencio,European Journal of Physics, 16, 191--194 (1995). [PDF]
Classical Electrodynamics 3rd Edition Pdf
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=ct/n and r=ct/n, and n is the index of refraction of the medium, These singularities arise due to an idealizationof 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 entitledCherenkov 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]
X. Web pages of interest
1. One of the most prolific sources for pedagogical treatments ofa variety of interesting problems in classical electrodynamicsis a webpage maintained by Kirk T. McDonaldentitled Physics Examples and other Pedagogic Diversions.Although some of the content is password protected, most of thearticles listed are available freely.
2. Which direction is the electric field rotating in a left or right circularlypolarized wave? Check out the Wikipedia page on circularpolarization, which provides some enlightening animationsthat illustate the answer to this question.