PHYSICS 406

0.     Last semester we began the study of time-dependent phenomena, finally, in Ch. 7,
with discussions of Faraday's Law and Maxwell's (time-dependent) addition to Ampere's Law, which required us to understand the concepts of emf, mutual- and self-inductance,, induced electric and magnetic fields, and "displacement current."
We used this to extend our understanding of (potential) energy in electric fields to that also contained in magnetic fields.
One should review this material at the same time as we begin our study of conservation laws in Ch. 8

Schedule for the Chapters
1. Ch. 8 Conservation Laws: (4 lectures), 22-31 August.

   We begin with the continuity equation, connecting spatial derivatives of current and time derivatives of charge density. This is an important consequence of Maxwell's equations. [Section 8.1.1]

   We introduce the important concept of the Poynting vector, which describes the time rate of change of momentum density in the electromagnetic field, recalling the notions of energy density that we have already studied. [Section 8.1.2.]
   We will talk about a "mystery" or two with respect to this, taken from the Feynman Lectures, Vol. 2.

   This chapter spends considerable effort describing the stress-energy tensor for electromagnetism. We will definitely talk some about what it says concerning the change per unit time per unit volume of

   • energy,
   • and momentum,

   as well as associated ways of using them to measure the total energy and momentum, all of which are quite important notions.

2. Ch. 9 Electromagnetic Waves: (6 lectures), 5-21 September

   This chapter begins by describing one-dimensional wave equations for strings, Sections 9.1.1-3. I assume that you learned these notions back in Phys. 160, and again in Phys. 303/4, and have only forgotten about half of it by now. Therefore we will begin with the 3-dimensional versions for electromagnetism, which are given in Sections 9.2.1-2. This material mainly considers monochromatic waves, i.e., those with a single frequency, also referred to as harmonic waves. One then uses these waves to create more general waves as sums of monochromatic wave solutions; the methods to do this are referred to as Fourier transforms. There are some useful, additional notes on such the properties, and uses, of these transforms on the main webpage.
   We will also make some "retreat" back to Section 9.1.4 on the polarization of wave fields, and will discuss polarization in some little more detail than is given in the text: an appropriate reference on Circular polarization is on the main webpage

   Then we discuss how the energy- and momentum-densities are transferred by electromagnetic waves, recalling the Poynting vector. [Section 9.2.3.]

3. The First Exam will be on Tuesday, 19 September. It will cover Chapter 8 and Sections 1 and 2 of Chapter 9, on general wave motion, as well as handouts on Fourier transforms and circular polarization.

4. The continuation of Chapter 9:

   We discuss the behavior of radiation within (linear) media, and use this to consider what happens when the wave moves from one medium to another:

   • reflection, transmission, refraction [Sections 9.3.]

   More interesting is the behavior when one, or both, of the media are conducting! Several different situations will be considered, including those in Sections 9.4.

   We will be very brief, if at all, in a discussion of rectangular wave guides, and/or coaxial transmission lines. [Sections 9.5.]

5. Ch. 10, Section 10.1 General E&M Potentials: (2 lectures) 26, 28 September.

   The time-dependent version of potentials for the electric field. We discuss these, the ambiguities in their definitions, and various "preferred" ways to resolve those ambiguities---gauge transformations.

6. Skip to Chapter 12, Section 12.1: (3 lectures) 3, 5, 10 October.

   In Phys. 262 and in Phys. 330, you learned about time dilation, length contraction, and the impossibility of the Newtonian ideas concerning simultaneity. We review those briefly, including the so-called "twin paradox," and then spend some time on the geometry of Minkowski space, the 4-dimensional spacetime in which special relativity "lives."
   We also learn to work with Minkowski diagrams, and talk a little about the allowed coordinate transformations in Minkowski space, namely 3-dimensional rotations, and Lorentz boosts.
   An especially interesting notion is the concept of a uniformly accelerated observer, which we will discuss, and for which there is, again, an online set of additional materials.
   All this is needed to properly discuss the next sections of Ch. 10

7. Fall Break comes next, with some time to think up questions about all this unusual material, i.e., special relativity and Minkowski space.

8. The Second Exam will be on Tuesday, 24 October. It will cover the remainder of Chapter 9, and also Sections 1 and 2 of Chapter 12.

9. The continuation of Chapter 10: (4 lectures) 17, 19, 24, 26 October.

   We spend enough time discussing the concept of retarded time, and introduce the retarded potentials. [Section 10.2.1.]

   We will skip Section 10.2.2, on Jefimenko's work.

   The remainder of the chapter, Sections 10.3, are devoted to the full answer to the questions as to the behavior of the potentials and the fields for a point charge moving in any arbitrary way, including acceleration!

10. Ch. 11 Radiation (6 lectures), 31 October, 2, 7, 9, 14, 16 November.

    We begin with the general notions of radiation: that it is the part of the electromagnetic field that sends non-trivial amounts of power to infinity. [Section 11.1.1.]

    We then look at the special cases of the electric and magnetic dipoles, [Sections 11.1.2-3.]

    and then the general case as described in Section 11.1.4; however, we will append some extra thoughts about quadrupole radiation, and how it is distinctly different from the dipole.

    Then we will append some more discussion about the relevance of all this to actual antennas, taken from a different source!

    Lastly we will apply all this to the power radiated by accelerating point charges. [Section 11.2.1.]

    We will skip the material on radiation reaction, Sections 11.2.2-3.

11. The Thanksgiving holidays begin on Thursday, 23 November.

12. Ch. 12 Special Relativity, applied to Mechanics and Electromagnetism: ( lectures) 21, 28, 30 November, 5 December

    Having already spent some time on the basic (4-dimensional, Minkowski) geometry underlying special relativity, we will first talk about the general form of Lorentz transformations [Section 12.1.3 and some additional material.]

    We will consider transformation equations for various physical objects, including velocity, momentum, acceleration, and force.

    Then your text spends more time than we will have time for, discussing mechanical problems in relativity. We will spend a little time on Sections 12.2.2-5.

    However, what we really want is the material in Sections 12.3, on the transformations of the electromagnetic quantities: the fields, the sources, the potentials, as well as the configuration of the fields into a skew-symmetric, second-rank tensor, F_mn. We hope to mention the Hertz potential as well.

13. The Third Exam will be on Thursday, 7 December. It will cover all remaining material.
14. There will NOT be a Final Examination.

• all the important physical quantities are vector fields, depending on 4 different coordinates, x,y,z,t.
• the relevant equations involve partial derivatives, and partial differential equations,
• applications usually involve complicated boundary shapes, thereby introducing functions adapted to those shapes.

Most of physics involves observations of phenomena at some (field) point, say , caused by occurrences at some (source) point, denoted by . Therefore it is very common to be interested in the difference of these two locations.
As you know Griffiths' text uses the symbol     to denote - , i.e., the vector displacement from the source point to the field point.
I find this symbol very difficult to create, or discriminate, both on the blackboard and on the computer monitor.
Therefore, I will always use the Greek symbol, to denote this difference! That is, I will use

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