# electrodynamics

"I have also a paper afloat, with an electromagnetic theory of light, which, till I am convinced of the contrary, I hold to be great guns."

Welcome to Physics 351! In this class we will study charges, currents, electric and magnetic fields, and their interactions. Much of the physics is expressed in a single, remarkable set of equations

\begin{gather} \vec{\nabla} \cdot \vec{E} = \frac{1}{\epsilon_{0}} \rho \vphantom{\frac{\partial\vec{B}}{\partial t}} \\ \vec{\nabla} \times \vec{E} + \frac{\partial\,\vec{B}}{\partial \,t} = 0 \\ \vec{\nabla} \cdot \vec{B} = 0 \vphantom{\frac{\partial\vec{B}}{\partial t}}\\ \vec{\nabla} \times \vec{B} - \mu_{0}\,\epsilon_{0}\,\frac{\partial\,\vec{E}}{\partial\,t} = \mu_{0}\,\vec{J} \end{gather}

This formulation of electromagnetism is due primarily to the Scottish physicist James Clerk Maxwell. His equations, in one form or another, describe phenomenon ranging from the propagation of light to the deflection of a compass needle by a magnetic field.

James Clerk Maxwell (1831-1879)

The impact of Maxwell's equations extends well beyond electromagnetism: the Theory of Special Relativity is secreted away inside them, and they are the prototype for a unified description of the basic forces of Nature.

## Syllabus

Basic information about our schedule, homework assignments, grades, and more can be found below. Click here to download a pdf version of the full syllabus. The syllabus has more detailed information, and you should be familiar with the policies and rules it describes.

## Fall 2022 Schedule

We will cover most of the first seven chapters of the textbook, along with parts of chapters 8 and 9. The table below is an estimate of how we'll spend our time.

Week Dates Chapter
1 August 30; September 1, 3 1
2 September 6, 8, 10 Labor Day; 1, 2
3 September 13, 16, 18 2
4 September 20, 23, 25 2
5 September 27, 29; October 1 2, 3
6 October 4, 6, 8 3, Exam 1
7 October 11, 13, 15 Fall Break; 3
8 October 18, 20, 22 3, 4
9 October 25, 27, 29 4, 5
10 November 1, 3, 5 5
11 November 8, 10, 12 5, 6
12 November 15, 17, 19 6, Exam 2
13 November 22, 24, 26 7; Thanksgiving
14 November 29; December 1, 3 7, 8
15 December 6, 8, 10 9
16 December 18 Final Exam (1-3pm)

Please keep in mind that these dates are subject to change. We are coming back from remote learning (and who knows what will happen going forward) so there will probably be an adjustment period as we get used to being in-person again. I may move some things around as a result, or I might decide to spend more or less time on a particular topic. I will always notify you about any changes I make to this schedule.

## Assignments

Homework is assigned each week (except for exam weeks) and collected the following week. With the exception of the first few assignments it will usually be due on Monday at the beginning of class. That way you can ask questions during our Friday discussions.

Only some of the problems from each assignment will be graded. I won't tell you which ones, so you need to complete all the problems. We will talk more about how this works in class. Current and past assignments are listed below. Solutions are available for some (not all) problems, but they are not available for download. Please stop by my office if you'd like to see the solutions for a particular assignment.

Assignment 5
Electrostatic Potential Energy
Due on Monday, October 3

Homework 5 covers work and potential energy in electrostatics. There is also a bonus problem that investigates some common features of $$1/r^{2}$$ forces. That one won't be graded or checked for completion, but it's an interesting exercise!

Assignment 4
Electrostatic Potential
Due on Monday, September 26

This is the second homework for Chapter 2, covering the electrostatic potential, Gauss's Law, and the behavior of the electric field at surfaces where there is a charge density.

Assignment 3
Electrostatics
Due on Monday, September 19

This is the first homework for Chapter 2, covering Coulomb integrals and Gauss's Law. The rules about using Mathematica and similar tools are stated at the top of the assignment. (They are not allowed, just like on the last assignment.) Click here for some additional tips on problems 5 and 6. Some of the integrals you need are discussed in "A Few Useful Integrals," in the Notes section.

Assignment 2
More Vector Analysis
Due on Wednesday, September 7

This assignment covers the rest of our Math Methods review. Read the instructions at the top of the page – Mathematica and similar tools are not allowed!

Assignment 1
Review of Vector Analysis
Due on Wednesday, August 31

This assignment is due at the beginning of the second class. It is a review to get you up-to-speed on some aspects of vector analysis that we will frequently use in class.

Working with your classmates on these assignments is encouraged! But you should only hand in work that you've completed on your own. If your solution looks just like someone else's work then you need to go back and redo it from scratch. If you can't explain each step of your solution then you haven't completed the problem on your own. Remember: the only way to be ready for the exams is to do the homework yourself.

Warning

Never, ever hand in an assignment that has been copied from a solutions manual. You won't learn anything that way, and it will earn you a grade of zero for that assignment. If it happens more than once it will be reported to the Department Chair and the Dean. Consider yourself warned. Click here to see the College of Arts and Sciences Statement on Academic Integrity.

Grades in the course are primarily determined by homework assignments and exams. The weekly homework grades contribute 35% of your final grade in the class, and two exams (on October 8 and November 19) count 15% each. A cumulative final on Saturday, December 18 (from 1:0-3:00 PM) is worth 30%. The remaining 5% depends on attendance and participation. Asking questions, taking advantage of office hours, and attending both lectures and discussion sections will earn you the full 5%. Check the pdf syllabus for more details.

## References

The main text for the class is Introduction to Electrodynamics (4th Ed) by Griffiths. The tone of the book is casual and most students find it very accessible. When I was an undergraduate I used the the books by Wangsness and Purcell. Those texts might be useful if something in Griffiths isn't clear. A more advanced treatment is given in Jackson's Classical Electrodynamics, which is the text for practically every graduate E&M course.

1. Introduction to Electrodynamics
David J. Griffiths
2. Electromagnetic Fields
Roald K. Wangsness
3. Electricity and Magnetism
Edward M. Purcell
4. Classical Electrodynamics
J.D. Jackson

Griffiths' book has a very complete (for our purposes) discussion of vector calculus as it is used to describe electricity and magnetism. If you'd like to see additional discussions of this material, I recommend the math methods book by Boas, and also the book by Riley, Hobson, and Bence. For a more advanced treatment refer to Arfken and Weber.

1. Mathematical Methods in the Physical Sciences
Mary L. Boas
2. Mathematical Methods for Physics and Engineering
K.F. Riley, M.P. Hobson, and S.J. Bence
3. Mathematical Methods for Physicists
George Arfken and Hans Weber

The Feynman Lectures on Physics, which include a few nice discussions about some of the things we'll talk about in class, are available online. There should also be a copy of the lectures in Isaac & Al's.

From time to time I may supplement the material from the book with my own notes, which will be posted below.

## Lecture Notes

The full set of lecture notes is available on Sakai, organized in the "Lecture Notes" folder under the "Resources" tab. Click here to access the notes.

## Notes

A quick derivation of the result we talked about in class, showing why part of the electric field has a discontinuity at a charged surface.

One of the problems on Assignment 4 leads to an integral of the form \begin{gather} \int dx\,\sqrt{x^2 + \alpha^2} ~. \end{gather} Evaluating this integral requires the application of several different integration techniques, including changes of variables, trig substitutions, and the method of partial fractions.

These notes review two calculations of the electrostatic potential: one obtained by integrating $$d\vec{l} \cdot \vec{E}$$ for a known electric field, and the other by adding up the infinitesimal contributions \begin{gather} dV(\vec{r}, \vec{r}\,') = \frac{dq(\,\vec{r}\,')}{|\,\vec{r}-\vec{r}\,'|} \end{gather} to the potential from each bit of a distribution of charge.

These notes explain why the electric field has a scalar potential, and how to find it based on the distribution of charge. Please familiarize yourself with the results in these notes before Dr. Tangarife's lecture on Friday, September 14.

When a charge distribution is very symmetric, Gauss's Law can help us determine the electric field without having to set up and evaluate Coulomb integrals. These notes briefly review Gauss's Law, Gaussian surfaces, and how to find the electric field for a spherically symmetric distribution of charge.

There were some issues on homework 2 with how to evaluate integrals containing three-dimensional Dirac deltas. Here are a few examples you can work through if you'd like a little more review.

There is an integral requiring a trig substitution that shows up a few places on Homework 3. If you are stuck you should read these notes for an explanation.

Some extra discussion of charge distributions, the transition from a collection of point charges to an infinite number of infinitesimal charges, and the Coulomb integrals for the electric field produced by line, surface, and volume charge densities.

These notes give a brief overview of the Helmholtz theory of vectors, and some important facts about vectors with vanishing divergence or curl. A more complete discussion is given in Appendix B of the text. Some of these ideas will be developed more fully in later chapters.

A quick review of a few integrals that show up again and again on the homework.

The Dirac delta can be a little tricky, so here are some notes that expand on our quick review in class.

A very quick and very rough review of line, surface, and volume integrals with several examples. The part on volume integrals isn't finished, but the stuff on line and surface integrals is there.

A review of orthogonal coordinate systems and vector calculus for students who did not take Phys 301 (Math Methods) in the Spring 2020 semester.

This is a very basic review of line integrals – what they are, how to evaluate them, etc. It may be useful if you're a little rusty on this topic. The file is big (about 22 MB) because of the various plots. Let me know if you find any typos or mistakes and I will post a corrected version.

## E&M Stress Relief

Sometimes the E&M wears you out, and you need a picture of an adorable little kid doing physics to get you back on track. Not a problem.