Relativistic Electrodynamics

Relativistic Electrodynamics

Prof. Erich Poppitz

office: 1113A, 416-946-7546

email: poppitz physics utoronto ca

Syllabus and literature

This course illustrates, using classical electromagnetism, how symmetry principles and scaling arguments combine to determine the basic laws of physics. It is shown that the electromagnetic action (from which follow the equations of motion) is uniquely fixed by the principles of special relativity, gauge invariance, and locality. Additional topics include motion of relativistic particles in external electric and magnetic fields, radiation from point charges, and the breakdown of classical electromagnetism.

Required - “The classical theory of fields” by L. Landau and I. Lifshitz.

Recommended, less advanced but with many problems: “Introduction to electrodynamics”

by D. Griffiths

As in PHY354, my motivation for this choice of required text is that it is the classic of the field---which is hard to beat in its completeness and depth (it is also thinner and hence easier on the wallet). I will be posting my class notes which will contain extended derivations of the covered material.

We will be using www.piazza.com for updates, posting of homework, solutions,

and online discussions. All registered students will receive an email inviting them to register.

The “competition”: MOOC’s? (I couldn’t find any on this topic).

However, the Stanford School of Continuing Studies has posted 10 lectures on “Special Relativity” by Leonard Susskind, one of the world’s most prominent theoretical physicists, who has made many seminal contribution in elementary particle theory. These lectures last a total of 1062 mins = 21.2 x 50 mins, i.e., almost as much as our class of 24 50 min lectures. Check them out on youtube - there is no playlist but search for “susskind special relativity” (note that I do not mean the “modern physics” series, where special relativity is also discussed). Same effect as that seen in the “Classical Mechanics” class by same professor - the number of views drops from 35k in first lecture to a steady 4-8k in the following ones.

NOTE: The notes below are left here for reference. They contain unfixed errors, incl. undiscovered ones.

You can also check out these lovely type-written notes by Peeter Joot, a very motivated student who took the class a few years back: http://peeterjoot.com/archives/math2011/phy450.pdf

Lecture notes - (current status of lectures: see www.piazza.com)

In order to orient you about the scope and level of this class, these are left online from last year. The order may change as we go on, but it is unlikely that we will have time to add more topics, despite the desire to do so.

For those who want the chapter numbers from the book, here they are: all of Ch. 1; Ch. 2 Sect. 8, 9; all of Ch. 3; Ch. 4, skip Sect. 34, 35; material of Ch. 5 has been studied in previous courses, assumed to be known; Ch. 6, Sect. 46, 47; skip Ch. 7; Ch. 8, Sect. 62, 63, 65; Ch. 9, Sect. 66, 67, 71, 73, 75. Wish we could do more but that’s the most possible with 24 lectures.

pp.1-11: The Gallilean principle or relativity (1-6). Reasons that led to the necessity to reconsider it (7-9). Einstein’s generalization (10) and relativity of simultaneity (11).

Some scribbles intending to address two questions raised in/after class on Wed. Jan. 11: one regarding the Michelson-Morley experiment and another about simultaneity. Here’s a brief summary: 1.) On pp.1-3, I give a calculation of the time difference between the two paths in the interferometer for the case when the platform’s motion w.r.t. the aether is along one of its arms. You see that turning the interferometer on 90 degrees produces a sign difference in the phase mismatch, for equal arms lengths, but not turning it on 180 degrees. 2.) On p.4, I “reverse” the simultaneity argument of p.11 of notes, stressing the point that receiving something at a given point in space at a given time is an event - just like the “decay” or “explosion” in two parts is. I am providing these rather messy notes because I have no time to address the issue in depth in class - please ask me after class or during office hours if you want more explanations.

pp.12-26: The notions of spacetime, event, worldine, and the interval (12-14). The invariance of the interval between lightlike events and the invariance of arbitrary infinitesimal intervals as a consequence of constancy of “c” and spacetime symmetries (15-17). The geometry of spacetime: lightcone (18-19), timelike and spacelike regions (20-21). Proper time (22-24), finishing the proof of invariance for finite (timelike) intervals (25-26).

Here are some notes prompted by questions in class on Thur. Jan. 12: precise meaning of ds^2 (15.1); independence of A in ds^2 = A ds(prime)^2 on space or time separation (15.2); why A=-1 disallowed? (15.3-4). Finally, a less Landau-Lifshitz-like “baby” argument why s = s(prime) for finite timelike separated events (15.5-7)

Note that, accidentally (not!), this argument also shows time dilation-just compare the time differences between the emission and absorption of the light ray in the two frames (first eqn. on p. 15.5 and eqn. (*) on p. 15.6).

p. 27-44: analogy of Lorentz transformations to rotations in four-dimensional Euclidean space (27-29);

derivation of Lorentz transformations (30-31); spacetime diagrams in different frames, tachyons, and causality (32.1-32.3); nonrelativistic limit (33); number of parameters of the Lorentz group, three vectors and four-vectors (34-37); Minkowski space metric, Lorentz transformations, and index notation (38-44).

p. 45-51: importance of covariance of equations of motion (45-46); four-velocity and four-acceleration (47-49);

action for free relativistic particle (50-51).

p. 52-56: equation of motion from least action principle (52-54); conservation laws: energy and momentum (55-56);

p. 53.1, completing an incomplete argument.

p. 54.1-54.4: derivation of the equation of motion for a free relativistic particle from the geometric action, directly in four-vector form

p. 56.1-73: the kinds of external fields a particle can “feel” and the interaction lagrangian: Lorentz scalar field (59-62), Lorentz vector field, its transformation properties and interaction lagrangian (63-65); the equation of motion of a relativistic particle in an external electromagnetic field (66-68, 73); please read pp. 69-72 by yourself!

p. 74-83: gauge invariance; energy (74-75); EOMs in 4-vector form and the field strength tensor (76-80); energy conservation (81); transforms of F^{ij} and some math remarks (82-83).

p. 84-102: invariants build of F^{ij} (84-86); “Bianchi identity” and the first half of Maxwell’s equations (87-90); action for the field degrees of freedom (91-94); 4-current for a system of particles and its coupling to the 4-vector potential (95-102). PLEASE READ pp. 98-100 of notes (on delta functions) for Wednesday’s (March 1) class!

p. 103-113: variational principle for the four-vector potential, boundary conditions, and Maxwell’s equations (103-108); Coulomb gauge, the wave equation in vacuum, and the speed of light (109-112); the wave equation in Lorentz gauge (113).

p. 114-127: the wave equation in vacuum and its most general solution (114-121 - PLEASE READ the math interlude on Fourier transforms contained herein!; transversality and polarization of the electromagnetic waves (122-125); energy and momentum of the field and the Poynting vector (125-127)

GF1-5: Green’s functions of the Laplace and D’Alembert equations. SELF READ. See also HW4.

p. 128-135: Poynting vector, momentum and energy density of EM waves (128-129); experimental consequences of the momentum of the EM field and their relevance (130-132); equation with sources in Lorentz gauge (133-135)

Notes on parity and chirality, prompted by a question in class.

p. 136-146: reminder on Green’s functions of Laplace equation (136); the retarded Green’s function of the D’Alembert equation (137-140); solution of the 4-vector potential of a general source (141-142); the Lienard-Wiechert potentials (142-146).

p. 147-165: fields of moving charges - particle at rest (147), particle in uniform motion (148-152), particle along an arbitrary worldline, energy flux, and radiation (152-154); the notion of radiation zone and radiation wavelength (155-157); the multipole expansion for radiation (158-159); electric dipole radiation and its power (160-164).

p. 166-180: quick dissussion of magnetic dipole (166; a more thorough discussion by Simon in tutorial on March 22); energy and momentum density of the EM field as element of a covariant object---energy-momentum conservation, spacetime symmetries of the EM field action, and the energy-momentum tensor (167-178); a simple example of its use in calculating EM field pressure (179-180).

p. 181-195: finishing up energy-momentum tensor discussion (180); the energy of a system of charged particles in the (v/c) expansion: leading order, (v/c)^0 (181-189); next-to-leading order (v/c)^2 and the ``Darwin Lagrangian (Hamiltonian)” (190-195 - these are largely superseded by the new notes below).

New notes for the last week (incorporating some pages from the the old ones): formally solving the coupled charge-field system and problems with the solution (1-4); the c->infinity limit (4.1); attempt to go next (v/c) order (5, 190-193,6); the Darwin Lagrangian (7) (comments on applications appear in the old notes on pp. 194,195); attempting (v/c)^3: a L&L derivation of the radiation reaction force (198.1,198.1); a ``baby” derivation (197-198) and limitations of classical E&M (200).