Electricity and Magnetism

"I have also a paper afloat, with a 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} \\ \vec{\nabla} \cdot \vec{B} = 0 \vphantom{\frac{\partial\vec{B}}{\partial t}}\\ \vec{\nabla} \times \vec{B} = \mu_{0}\,\vec{J} + \mu_{0}\,\epsilon_{0}\,\frac{\partial\,\vec{E}}{\partial\,t} \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.

Spring 2016 Schedule

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

Week Dates Chapter
1 January 19, 21 1
2 January 26, 28 1, 2
3 February 2, 4 2
4 February 9, 11 2
5 February 16, 18 2, 3
7 February 23, 25 3
8 March 1, 3 3
9 March 8, 10 Spring Break
10 March 15, 17 4
11 March 22, 24 4, Easter Break
12 March 29, 31 5
13 April 5, 7 5
14 April 12, 14 6
15 April 19, 21 7
15 April 26, 28 9

Please keep in mind that these dates are subject to change. I may decide to switch things around or spend more or less time on a given chapter. I will always notify you about any changes I make to this schedule.

Assignments

Homework is assigned each week and collected the following week. 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. Current and past assignments are listed below. Solutions are available for some (not all) problems, but I am no longer making them available for download — please stop by my office if you'd like to see the solutions for a particular assignment.

Assignment 12
The Poynting Vector, Electromagnetic Waves
Due on Friday, April 29.

The last homework covers a few topics from chapters 8 (the Poynting vector) and 9 (electromagnetic waves).

Assignment 11
The Vector Potential, Magnetization
Due on Friday, April 22.

This assignment covers calculations of the vector potential (including the multipole expansion), and magnetization.

Assignment 10
Magnetostatics, Biot-Savart Law, and Ampère's Law
Due on Friday, April 8.

The Biot-Savart law gives a straightforward way of calculating the magnetic field due to a steady current. In some cases it can lead to difficult integrals. When there is sufficient symmetry we can instead use Ampère's law to determine the magnetic field.

Assignment 9
Electric Fields in Matter
Due on Friday, April 1

This assignment covers dipoles, polarization, and the response of linear dielectric materials to electric fields..

Assignment 8
The Multipole Expansion
Due on Thursday, March 24

This is the last assignment for Chapter 3, covering the sections on the Multipole Expansion of the potential.

Assignment 7
Separation of Variables
Due on March 18

Separation of variables is an important technique for solving the Laplace and Poisson equations. It is often dramatically easier than evaluating the integrals for \(V\) and \(\vec{E}\).

Practice
Post-Exam Problems
Not Due

Here are two practice problems that target common mistakes from the first exam. Try to work through them the same way you would on an exam — without your notes, but with the formula sheet I gave you.

Assignment 6
Method of Images
Due on March 4

This is the first homework for Chapter 3, with problems that address the “method of images”.

Assignment 5
Electrostatic Potential Energy
Due on February 19

Homework 5 covers electrostatic potential energy, as well as some common features of \(1/r^{2}\) forces.

Assignment 4
Electrostatic Potential
Due on February 12

This is the second homework for Chapter 2, covering the electrostatic potential. Notice the question at the end of problem 6, after you calculate the potential.

Assignment 3
Electrostatics
Due on February 5

Correction: The figure in problem 6 gives the wrong charge density. Please use \(\rho=k/r^2\) as stated in the text of the problem. (Thanks Caity!)

This is the first homework for Chapter 2. 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.)

Assignment 2
More Vector Analysis
Due on January 28

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 January 19

This assignment is due at the beginning of the first 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.

A 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

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 “midterm” exams (dates TBA) count 15% each. A cumulative final on Friday, December 11 (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 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

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

Notes

Fields for Moving Point Charges
Ever wonder what the \(\vec{E}\) and \(\vec{B}\) fields produced by a moving charge look like? This (dense) set of notes solves Maxwell's equations — rewritten in terms of the potentials \(V\) and \(\vec{A}\) — for a moving point charge.

E&M with Mathematica
Over the course of the semester I've been pretty strict about when you can and cannot use Mathematica. For the most part you've used it to evaluate integrals, or to take care of basic (though tedious) vector calculus operations. To get an idea of what Mathematica can really do, check out the following links:

“On the Importance of Being Edgy”
“3D Charges and Configurations with Sharp Edges”

These blog posts by Michael Trott (a Senior Researcher at Wolfram) explore a wide range of problems in electrostatics and magnetostatics. Trott uses Mathematica -- really uses it -- to perform calculations and produce visualizations that would take us days or weeks using pencil and paper. If you have Mathematica installed you can download the articles and play around with the various calculations. Even if you don't have Mathematica on your computer, you can still download Wolfram's CDF Player to view interactive results in a browser.

Where are the Magnetic Monopoles?
The link in the title will take you to the arXiv page for the article “Introduction to Magnetic Monopoles”, by Dr. Arttu Rajantie. In class we stated that magnetic monopoles don't seem to exist in nature. If you're curious about that statement, this article may be of interest to you. Dr. Rajantie is a Reader in Theoretical Physics at Imperial College in London (the academic rank of “Reader” at a British university is roughly equivalent to “Professor” at an American university).

Separation of Variables for a Spherical Shell with Surface Charge
These notes provide a detailed discussion of an example we worked out in class: the potential inside and outside a spherical shell with the azimuthally symmetric surface charge density \(\sigma(\theta) = \sigma_{0} \cos\theta\). Please take a look, especially if you have questions about Assignment 7. (Also, please ignore the date referenced in the note -- they were typed up last fall.)

A Tricky Integral
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.

A Few Useful Integrals
A quick review of how to evaluate a few integrals that show up again and again on the homework.

Line Integrals

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.