Electromagnetic Field Theory III

Physics 836, Spring 2009

Welcome to the Physics 836 home page! The course information is available here plus lots of supplementary info. Please check this page regularly.

Office: M2046 Physics Research Building (PRB)

Office Hours: Mondays, Tuesdays, 11:00 am - 12:00 pm, or by appointment

Course Meets: Mondays, Wednesdays 8:30 am - 10:18 am, Smith 1042. (Exceptions will be announced in class and posted here.)

Grader: Ke Li (office: PRB 1112; phone: 247-8226; email: li.744@osu.edu)

Recent additions to this page:

*** By popular demand, the homework due date for HW#6 has been shifted to Thursday, May 28, 2009, 11:59pm. ***


*** The final exam will take place on Monday, June 8, 2009, 8:00-10:30am in PRB 1080 (Physics Research Building) ***
*** Bring paper, pencil, and eraser. This is a closed book exam. You may bring 2 letter-size sheets of handwritten notes ("cheat sheets"). ***
*** Most potentially useful formulae will be given in the problem statements for the exam. ***

*** First class meets Monday, April 6, 2009, 8:30am ***

Topics:

• Special relativity
• Relativistic charges in electromagnetic fields
• Radiation by moving charges, radiation damping
• Energy loss, Cherenkov radiation, bremsstrahlung, method of virtual quanta (if time permits)

Textbook:

• J. D. Jackson - Classical Electrodynamics, 3rd Edition (John Wiley & Sons, ISBN 978-0-471-30932-1, $95.95)

Recommended Reading :

• L.D. Landau, E.M. Lifshitz - The Classical Theory of Fields: Volume 2
• L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii - Electrodynamics of Continuous Media: Volume 8
• D.J. Griffiths - Introduction to Electrodynamics (this is a useful book to refresh your memory, but it does not cover all the material we will discuss in class)
• Hans C. Ohanian - Classical Electrodynamics, 2nd Edition (this has nice introductory chapters and practice problems on vector calculus and special relativity, but does not cover everything we will discuss in class)

Lecture Notes:

• Chapter 11: Special relativity
  • Lecture 1 (4/6/09)
• Lecture 2 (4/8/09)
• Lecture 3 (4/13/09)
• Lecture 4 (4/15/09)
• Lecture 5 (4/20/09)
• Chapter 12: Relativistic charges in electromagnetic fields
  • Lecture 6 (4/22/09)
• Lecture 7 (4/27/09)
• Midterm review (4/28/09)
• Lecture 8 (5/6/09)
• Chapters 14, 16: Radiation by moving charges, radiation damping
  • Lecture 9 (5/11/09)
• Lecture 10 (5/13/09)
• Lecture 11 (5/18/09)
• Lecture 12 (5/20/09)
• Lecture 13 (5/27/09)
• Chapter 13: Energy loss in a medium, Cherenkov radiation
  • Lecture 14 (6/1/09)
• Copy of a slide showing the Bethe-Bloch curves for particles with different masses as functions of momentum (6/1/09)
• Final exam review sheets (6/1/09)

Homework Assignments:

Homeworks are due at 11:59 pm on the due date (typically a Wednesday). Please put them in the grader's mailbox in PRB or slide them under his office door if the main office is already closed. Late submissions can be handed in until Thursday 5:00pm (the day following the due date), with an automatic deduction (late submission penalty) of 10 points.
(Solutions are password protected, they are for the use of OSU students and faculty only, please write to me if you belong to that group and are interested in accessing them.)
Each problem is worth 10 pts unless stated otherwise.

• HW 1 (due Wednesday, April 15, 2009) Jackson Problems 11.5, 11.6, 11.20, 11.23, and 11.24
  Hints: 11.5: Start from the inverse transformation to (11.19) and take differentials. Sort out parallel and perpendicular components at the end.
  11.6: Use the results from problem 11.5 to compute the rocket's speed as a function of Earth time, then integrate for 20 years in rocket time.
  11.20: Note that Jackson uses "particle physics units" for energy, mass, and momentum which are equivalent to setting c=1. For part (b), make appropriate approximations by first estimating the maximum opening angle before calculating it (again approximately, to 1% accuracy). You should find about 5 degrees. -- Solution 1
• HW 2 (due Wednesday, April 22, 2009) Jackson Problems 11.13, 11.14(a,b), 11.15, 11.17(a), and 11.18 (50 pts. total)
  Hints: Problem 11.14(a): It helps to write the expressions to be evaluated as traces of some matrix. This reduces the number of terms to be evaluated.
  Problem 11.15: (a) How do you check for parallelism? (b) Taylor-expand for small angles around zero or pi/2, keeping enough terms to get all leading-order contributions to E and B.
  Problem 11.17(a): This is similar to what we did in class when you work out the Lorentz-scalars and vectors in the K and K' frames.
  Problem 11.18: (a) Start from E', B' in the charge rest frame K'; compute E, B in the K frame, keeping only leading tems in the limit v -> c. Check that things look like a delta-function, then normalize the delta-function appropriately. (c) In covariant notation, a gauge transformation is A'_\mu(x) = A_\mu(x) - \partial_\mu \Lambda(x). -- Solution 2
• HW 3 (due Wednesday, April 29, 2009) Jackson Problems 11.27, 12.1, 12.2, 12.3, 12.5(b).
  Hints: Problem 11.27: (a) Remember j --> j/c when going from IS to Gaussian units.
  Problem 12.1: (a) Similar as Section 12.1B, but with a different choice for the Lorentz-scalar Lagrangian. (b) "Covariant form": see (12.35) for an example. "Space-time form": Write things out in terms of energy E and 3-momentum P.
  Problem 12.3: (a) Remember v = p/E (in units where c=1). Compute dx/dt = v and integrate it for x(t). (b) Try expressing x(t), z(t) as functions of E(t) (the particle's energy): x(E(t)), z(E(t)). Then eliminate E(t).
  Problem 12.5 (b): Using results from Problem 11.6, express things in terms of the proper time of the particle. This facilitates the Lorentz boosts (why?). -- Solution 3
• HW 4 (due Wednesday, May 13, 2009) Jackson Problems 12.9 (20 pts.), 12.13 (20 pts.; ignore the last sentence "Compare with...."), 12.14, 12.16 (altogether 60 pts.).
  Hints: Problem 12.9: (a) Write down the field of a magnetic dipole and solve for lines of constant magnetic force. Orient your c.o.s. conveniently in azimuthal direction. (b) Assume the particle moves in the equatorial plane. The problem states that the particle drifts in azimuthal direction. Why would it do so? See Sec. 12.4 (especially the first part related to "gradient drift", and Eq. (12.55) which you may use without deriving it). (c) As the particle moves in polar (+/-z) direction away from the equatorial plane, the magnetic field increases (the magnetic field lines move closer together). This is an example of the "magnetic bottle" effect discussed in class. See discussion around Eqs. (12.70)-(12.75) for help to solve this problem.
  Problem 12.13: You can get a lot of help on this (but perhaps not within a week!) by going to the library and looking at the "Bethe bible" (referenced at the bottom of p. 598). Start by writing down the free and interacting parts of Eq. (12.82) for the case of 2 charged particles. (a) To transform from individual particle coordinates and momenta (velocities) to relative and center of mass coordinates, you can use the non-relativistic expressions since the (first-order) relativistic corrections are already written out exp[licitly in the Darwin Lagrangian. The main point of part (a) is to rewrite the Darwin Lagrangian in relative coordinates ("Bethe-Salpeter form") before computing the corresponding canonical momenta. You should find p = (m1 m2)/(m1 + m2) v + (1/2) [m1 m2 (m1^3+m2^3)]/(m1+m2)^4 (v^2/c^2) v - [(q1 q2)/(c^2 r)] [(m1 m2)/(m1+m2)^2][v + v_r e_r]. (c) This part requires eliminating v in terms of p by expanding in 1/c^2 and keeping only the 0th- and 1st-order terms.
  Problem 12.14: Is this Lagrangian gauge invariant? The answer to this question is a hint for part (a).
  Problem 12.16: The Proca Lagrangian (12.91) describes massive photons (see discussion in Sect. 12.8, especially Eq. (12.95) which is the relativistic energy-momentum relation for massive particles if you use p = hbar k, E = hbar omega, etc.). The rest of the problem requires you to simply go through the same steps again as we did in class for Maxwell (i.e. massless photon) fields. -- Solution 4
• HW 5 (due Wednesday, May 20, 2009) Jackson Problems 14.4(b), 14.5, 14.9, 14.10 (20 pts.), and 14.12 (60 pts. total) -- Solution 5
• HW 6 (due Thursday, May 28) Jackson Problems 14.8, 14.13, 14.16, 14.21, 14.26 (20 pts.) [60 pts. total]
  Hints: Problem 14.8: Write down the relativistic Larmor formula for the radiated power and compute dp^\mu/d\tau in the frame of the fixed charge. Then integrate the power along the particle's trajectory.
  Problem 14.13: Use the periodicity of the orbit to write down a discrete Fourier series for the integrand in (14.67). Plugging it into (14.67) you should be able to show that dI/(d\omega d\Omega) has a discrete frequency spectrum. Integrate over frequency and write dI/d\Omega = \int dt dP/d\Omega. Average dP/d\Omega over the orbit, [dP/d\Omega]_orbit, and identify the contribution from each mode m\omega_0, dP_m/d\Omega, to get the desired result.
  Problem 14.16: Use that energy is carried away by photons of energy \hbar\omega, and express \vec k in spherical coordinates, to derive k^0 d3N/d3k = c^2/(\hbar\omaega^2) dI/(d\omega d\Omega). Follow the hint in the problem to derive the given expressions for \rho^2 and \xi. Starting from (14.79), derive an expression for \hbar\omega d3N/d3k and compare it with the desired expression given in the problem. Show that the coefficients of the modified Bessel functions agree. This requires you to go once more through the derivation of (14.79) presented in class. Note that, in the chosen coordinate system, the polarization 4-vectors are purely space-like and have zero time components. Relate \epsilon_1,2 in the problem to \epsilon_\parallel,\perp used in class and the textbook derivation of (14.79).
  Problem 14.21: The Bohr model of hydrogen-like atoms gives r_n=an^2/Z, with Bohr radius a=\hbar^2/(m e^2), for the radius of it's nth state, and E_n = T_n + V_n = -(1/2) Z e^2/r_n for its energy. The decay time \tau is calculated by asking how long it takes to radiate classically the energy \Delta E = E_n - E_{n-1}.
  Problem 14.26: (a) Use (12.42). (b) The power spectrum is defined as P(\omega,E) = (1/T) dI/d\omega (i.e. the energy radiated per orbit). Use (14.89/90) to evaluate it for small and large frequencies and study how in these limits the power spectrum scales with frequency \omega and particle energy E. (c) Average the power spectrum with the energy distribution N(E) to get P(\omega). (d) What does the reported change in the index \alpha of the synchrotron radiation spectrum tell you? To work out the electron energy E in terms of \omega_c it may be useful to express things in terms of the "electron cyclotron frequency per field" e/(m_e c) = 1.76 x 10^7/(gauss sec) (in Gaussian units) and the "classical electron radius" e^2/(m c^2) = 2.82 x 10^{-13} cm. (e) Check Wikipedia (or similar) for information about the Crab nebula. How and when was it created? What other objects are found in the area of the nebula? -- Solution 6
• HW 7 (due Wednesday, June 3, 2009, 11:59pm -- hard deadline!) Jackson Problems 16.1, 16.2, 16.3, 16.7, 16.9(a).
  This is the last HW set this quarter. It consists of 5 problems worth 50 pts. total (the last problem (16.9(a)) is really easy for a change!).
  Hints: Problem 16.1: All I can say is: Virial Theorem!
  Problem 16.2: (a) Again, virial theorem! Express the energy and angular momentum through the radius of the orbit. (b) Consider the quantum numbere n as a continuous variable and identify -dn/dt with the transition probability 1/\tau considered earlier in Problem 14.21. (c) Here are some useful combinations of natural constants in CGS units: e^2/\hbar c = 1/137, \hbar c = 197.33 MeV fm, m_\mu c^2 = 105.66 MeV.
  Problem 16.3: (a) Remember Kepler's laws, especially the second and third! (b) Divide the two rate equations by each other and solve for \epsilon(L). Plug the result into the expression for the eccentricity. (c) How are binding energy and angular momentum related for a circular orbit? Do the equation in (a), (b) preserve this relation in time (i.e. do circular orbits remain circular?)? Express for circular orbits \epsilon and L by the radius r of the orbit and show that the equations in part (a) reproduce the equation in Problem 16.2(a).
  Problem 16.7: Consider the first and second (proper) time derivatives of p^\mu p_\mu. Show that F_rad^\mu has the correct non-relativistic limit and satisfies all properties of a 4-force.
  Problem 16.9(a): Easy if you use what you derived in Problem 16.7. -- Solution 7

Exams:
Midterm Exam: Monday, May 4, 2009, 8:00-10:18am in Smith 1042 (please note early start!)
Final Exam: Monday, June 8, 2009, 8:00-10:30am in PRB 1080 (please note non-standard room!)

Grading: 40% HW, 30% Midterm, 30% Final