# Physics 836: Classical Electrodynamics (Spring, 2001)

#### Introduction and General Format

Physics 836 is the third quarter of a full-year sequence on Classical Electrodynamics. The text will continue to be Classical Electrodynamics,'' by J. D. Jackson (3rd Edition). However, I expect to deviate significantly from the material in this book during the latter part of the quarter. The course will meet MWF from 9:25 to 10:25 in Smith 2186, with (rare) meetings Friday at 8:30. The instructor is David Stroud.

You can choose to be graded in either of two ways. Method I: Homework and final only. If you choose this method, your grade will be based on the problem sets (about 40%) and a final (about 60%). Method II: Homework, paper, and final. If you choose this method, the grade will be based on problem sets (about 25%), a final (about 35%) and a paper (about 40%). The paper can be either a written paper, an oral presentation, or a computer project. Possible topics will be announced in a few days, and I will also consider your own choices. You can also work in teams of two, with one person giving a written presentation and one a talk. There will not be a midterm this quarter.

The final exam will be given on Wednesday, June 6 from 9:30 - 11:30 in our usual classroom. It will be open book (that is, you may bring any or all of Jackson, a book of math tables, your class notes, and homework solutions). The exam will be comprehensive - that is, it will cover topics from any part of Physics 836.

If you have not already given me your email address, please do so in class or by email (stroud@ohpsty.mps.ohio-state.edu). I plan to send problem sets and other announcements by email.

#### Syllabus

I expect to cover Special Relativity (i. e., roughly, the material in Jackson, Ch. 11 and 12) early in the quarter. Beyond that, I will discuss a range of special topics. These will be selected from the following: radiation from moving charges; Cherenkov radiation; photonic band gap materials and electromagnetic waves propagating in periodic media; X-ray diffraction from solids; inhomogeneous dielectric media and percolation theory. (I don't expect to cover all of these.)

#### Problem Sets

I expect to have problem sets as in previous quarters, approximately weekly but probably slightly less frequently. Unless otherwise specified, they are due each Wednesday by 5PM in my mailbox (or preferably, the grader's mailbox) in the physics office (or you can turn them in at classtime, of course). Unless otherwise stated, each problem is worth 10 points.

#### Problem Set 1

Due Wednesday, April 4 by 5PM

1. Jackson, Problem 11.3.

2. Jackson, Problem 11.6.

3. Jackson, Problem 11.8 (a).

4. Jackson, Problem 11.5. [In this problem, prove only the second identity, since the first was derived in class. You can start from eq. (11.31).]

#### Problem Set 2

Due Wednesday, April 11 by 5PM

1. Write out Jackson (11.141) and (11.143) and show that they give the Maxwell equations (i. e. the eight scalar equations which make up the Maxwell equations).

2. Jackson, problem 11.15

3. Jackson 11.14 (a), (b).

4. Consider an infinitely long wire of charge \lambda per unit length (in its rest frame). The wire moves with speed v parallel to its length, relative to the lab frame K

Calculate the magnetic induction B in the lab frame in two ways:

(a) Find the current density in the lab frame from a Lorentz transform, and and use Ampere's Law.

(b). Find the electric field in the rest frame of the wire and Lorentz-transform the fields.

Show that both methods give the same answer.

5.Write out the components of Jackson, eq. (11.144) explicitly and show that they give the Lorentz force equation and the corresponding power equation as discussed in class.

#### Problem Set 3

Due Wednesday, April 18

1. Jackson 11.22(a).

2. Jackson, 12.3

3. Jackson 12.5(b).

4. Starting from the Lagrangian (12.12) for a charged particle in a specified electromagnetic field, fill in the steps to obtain Jackson's equations (12.16) and (12.17).

#### Problem Set 4

Due Wednesday, April 25 by 5PM

1.Jackson 11.21(a).

2. As we showed in class, in a large electromagnetic cavity of volume V = abc, the allowed k points are (k_x, k_y, k_z), where k_x = 2\pi/a, k_y = 2\pi/b, and k_z = 2\pi/c.

(a). Suppose for simplicity that all three box edges are of length L. Find the density of modes per unit volume in k-space, recalling that there are two modes for each k point.

(b). Calculate the number of modes between angular frequency \omega and $\omega + d\omega, in the limit of very large L. (c). Suppose that the total energy in a mode of frequency$\omega$at temperature T is \hbar\omega/[exp(\hbar\omega/k_BT) - 1], where \hbar is Planck's constant. Let E(\omega)d\omega equal the energy in the cavity between frequency \omega and \omega + d\omega. Find the frequency for which E(\omega) is maximum at a given temperature. Estimate this frequency for (a) room temperature, and (b) the surface temperature of the sun. 3. Jackson 11.27 (a). #### Problem Set 5 Due Wednesday, May 2 5PM 1. (a). Fill in the steps needed to go from Jackson, eq. (14.25) to Jackson, eq. (14.26). 2. Verify eq. (14.66). 3. Jackson, problem (14.4). Here, the word discuss'' means calculate.'' 4. Jackson, problem (14.5) 5. (Optional) Repeat the calculation done in class for the solution to the wave equation for a blip source,'' but carry out a contour integral suitable for the advanced solution rather than the retarded solution, and obtain the advanced solution. #### Problem Set 6 Due Wednesday, May 9 by 5PM 1. Jackson, problem (14.9). (a), (b), (c) 2. (a) Consider the reciprocal lattice of a Bravais lattice. Show that the reciprocal of this reciprocal lattice is the original Bravais lattice. (b). Show that the reciprocal of a face-centered-cubic lattice is a body-centered cubic lattice. (c). Show that the reciprocal of a simple hexagonal lattice is also a simple hexagonal lattice, but rotated with respect to the original one. (Note: simple hexagonal'' means the same thing as what I called in class stacked triangular''). #### Problem Set 7 Due Wednesday, May 23 by 5PM 1. As stated in class, a diamond structure can be viewed as two interpenetrating face-centered-cubic lattices, one of which is displaced by vector b = (a/4, a/4, a/4)$ relative to the other, where a is the cube edge.

(a). Write down the basis vectors for the reciprocal lattice of the face centered cubic structure.

(b). Show that the X-ray scattering cross-section from silicon (in which the atoms are located on a diamond lattice) vanishes for certain reciprocal lattice vectors of the fcc lattice, and find those reciprocal lattice vectors.

(c). For which reciprocal lattice vectors do the scattered waves from the two sublattices interfere constructively?

2. Suppose that a given crystal has one atom per Bravais lattice point R, and that the total electron density has the form

n_e(x) = \sum_{\bf R}n_{at}(x- R),

where n_{at}(x) is the electron density associated with a given atom. As shown in class, the differential scattering cross-section at reciprocal lattice vector K is proportional to |n_e(K)|^2, where n_e(K) is the Fourier transform of the electron density.

Find an expression for |n_e(K)| in the case when n_{at}(x) is the electron density characteristic of the 1s state of hydrogen.

3. Consider a periodic layered magnetically permeable medium, with alternating layers of permeability \mu_1 and \mu_2, and thicknesses d_1 and d_2. The dielectric constant of each layer is unity. (We are using esu units here.) Consider a linearly polarized plane electromagnetic wave of frequency propagating in a direction perpendicular to the layers. The first layer of medium 1 occupies the region from z = 0 to z = d_1; the first layer of medium 2 occupies \$_1 < z < d_1 + d_2.

(a). Write down the most general form of the wave (i) for 0 < z < d_1; (ii) for d_1 < z < d_1 + d_2.

(b). Use Bloch's theorem, combined with the known boundary conditions at d_1 and d_1 + d_2, to write down four conditions on the four unknown amplitudes you found in part (a).

(c). Hence, write down a determinantal condition which determines the frequencies for a given Bloch vector k. Do not attempt to solve this condition.

#### Term Project

The optional term project for Physics 836 will consist of an oral or written presentation, or a computer project, on a special topic involving applications of electromagnetism to a problem of current research interest. The project should be designed as follows:

You do not need to do any original research yourself.

The term paper may consist of an oral or a written presentation, or a computer project.

If written, the paper should be of order 10-15 pages, TYPED and double-spaced. It should include suitable references and may also include a few carefully selected figures. Note: the equations DO NOT have to be typed.

The deadline for the paper is Monday, June 4 at 11 AM. No extensions will be allowed. I would very much prefer to receive the papers by Friday, June 1.

If an oral presentation, the talk should be about 30 minutes, of which about 5-8 should be left for discussion. The paper can be presented using transparencies (preferred) or as a chalk talk. If I can find a way to get a projector into the classroom (doubtful), I might consider a powerpoint presentation.

The paper will be graded on both presentation and content. By presentation,'' I mean clarity and organization of the exposition. By content,'' I mean how well you understand the chosen topic, as revealed in the paper or talk. In judging content, I will take into account the difficulty of the topic chosen - I do not expect the same degree of mastering of the most difficult topics, and I expect GREATER understanding of those topics which have already been partly covered in class (i. e., you will have to go beyond the classroom material). At the moment I expect to give about 40% for presentation and 60% for content.

A non-inclusive list of some possible topics is given below. I will be adding to these in the next few days, and I invite you to propose your own topics (but these must be approved by me).

I will consider allowing projects done in teams of two. In this case, one person will give the talk and one will write the paper. Such projects must be approved in advance.

The time table for these projects is given at the end of this assignment (subject to small changes).

Some possible term paper topics for Physics 836:}

Synchrotron radiation: frequency-distribution, applications to studies of solids, biology, etc.

Electromagnetic wave propagation through anisotropic materials: this includes several possible topics, including magneto-optics, optical rotation, optically active materials (such as biological molecules).

Problems in nonlinear optics: optical Kerr effect, second harmonic generation, self-focusing of electromagnetic waves.

Cherenkov radiation and its applications.

Topics in magnetohydrodynamics

Electrodynamics of superconductors (London equations, Ginzburg-Landau equations, Abrikosov flux lattice).

Models for the complex dielectric functions of real materials.

Problems in light scattering: Rayleigh scattering, Brillouin scattering, Raman scattering, critical opalescence, electromagnetic scattering from periodic arrays of scatterers.

Aspects of the Hall effect and magnetoresistance of materials.

Interactions between gravitation and electromagnetic radiation.

Quantization of the electromagnetic field.

Light scattering from rough surfaces

Medical applications of electromagnetic radiation.

Electrorheological and/or magnetorheological fluids

Left-handed materials: nature and applications

Percolation theory: network models and simulations (this would make a good computer project).

Photonic band gap materials (this will be covered in class, but would make a good, if challenging, computer project)

Optical traps, optical lattices, etc.

Note: further topics will be added in the next several days. Some of the topics listed above are actually groups of several topics.

Time Table:

By Monday, April 23: written proposal of topic due. (This does not commit you to that topic, nor does it commit you to do a project.)

By Friday, May 4: half-page outline plus two or three references are due. Also, you should choose a written or oral presentation or a computer project by that date.

Monday, May 21: written papers due at beginning of class. Also, oral presentations will begin on or about this date. N. B.: All students required to attend all oral presentations.

#### Office Hours; Grader

My office hours will be Monday from 4-5PM and Tuesdays and Wednesdays 11:30 A. M. to 12:30 PM, in Smith 4034. If you are unable to make the office hours, please email me and I will arrange an appointment. My office phone is (292)-8140, and I can easily be reached by email (stroud@ohstpy.mps.ohio-state.edu).

The grader will continue to be Masa-oki Kusunoki (masa@pacific.mps.ohio-state.edu).

#### Random Information

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