*"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 hidden 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 2024 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 26, 28, 30 | 1 |

2 | September 2, 4, 6 | Labor Day; 1, 2 |

3 | September 9, 11, 13 | 2 |

4 | September 16, 18, 20 | 2 |

5 | September 23, 25, 27 | 3 |

6 | September 30; October 2, 4 | 3, Exam 1 |

7 | October 7, 9, 11 | Fall Break; 3 |

8 | October 14, 16, 18 | 4 |

9 | October 21, 23, 25 | 4, 5 |

10 | October 28, 30; November 1 | 5 |

11 | November 4, 6, 8 | 5, 6 |

12 | November 11, 13, 15 | 6, Exam 2 |

13 | November 18, 20, 22 | 7 |

14 | November 25, 27, 29 | 8; Thanksgiving |

15 | December 2, 4, 6 | 9 |

16 | December 12 | Final Exam (1-3 pm) |

Please keep in mind that *these dates are subject to change*. This is the absolute minimum that we have to cover. But if you are engaged and active in class we can go faster and cover additional (and interesting!) material besides what is listed here.

## Assignments

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

Only some of the problems from each assignment will be graded. I won't tell you which ones, so you need to complete them all. We will talk more about how this works in class. Current and past assignments are listed below. You can see solutions 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 11

The Vector Potential, Magnetization

*Due on Monday, November 25*

This assignment covers calculations of the vector potential (including the multipole expansion), and the properties of materials with a permanent magnetization. Be sure to click the link in problem 2 to learn more about the Aharonov-Bohm effect!

Assignment 10

Magnetostatics, Biot-Savart, and Ampère's Law

*Due on Monday, November 11*

This is a chance to practice calculating the magnetic field using the Biot-Savart law and Ampère's Law.

Assignment 9

Electric Fields in Matter

*Due on Monday, November 4*

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

Assignment 8

The Multipole Expansion

*Due on Monday, October 28*

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 Monday, October 21*

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

Assignment 6

Method of Images

*Due on Monday, October 14*

Using the “method of images” to determine the electric field due to charges near an ideal conductor.

Assignment 5

Electrostatic Potential Energy

*Due on Monday, September 30*

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 problem won't be graded , but it's an interesting exercise!

Assignment 4

Electrostatic Potential

*Due on Monday, September 23*

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 16*

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. 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

Vector Calculus

*Due on Wednesday, September 4*

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 28*

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 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

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 4 and November 15) count 15% each. A cumulative final on Thursday, December 12 (from 1 - 3 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.

*Introduction to Electrodynamics*

David J. Griffiths

*Electromagnetic Fields*

Roald K. Wangsness

*Electricity and Magnetism*

Edward M. Purcell

*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.

*Mathematical Methods in the Physical Sciences*

Mary L. Boas

*Mathematical Methods for Physics and Engineering*

K.F. Riley, M.P. Hobson, and S.J. Bence

*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. The Physics Club should also have a copy downstairs.

From time to time I will supplement 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

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).

Fields and Lorentz Transformations

Two observers in different frames of reference will give different descriptions of charge and current densities. As we saw in class, this means they will also give different descriptions of the electric and magnetic fields produced by these sources. How does this work?

Why magnetic fields?

If all you know about is electrostatics and special relativity, you encounter paradoxes that can only be resolved by introducing a new field that exerts a force on moving charges.

Linear Dielectrics

A few comments about linear dielectrics, with an example of how to use Gauss's Law to find \(\vec{D}\) and then relate it to \(\vec{E}\) using \(\vec{D} = \epsilon_{0} \vec{E} + \vec{P}\).

Multipole Forces

Forces between multipoles look much different than the familiar Coulomb force between two point charges — they decrease more rapidly with distance and depend on the multipoles' relative orientations. In these notes we use what we know about the potential energy of pairs of charges and dipoles to work out monopole-dipole and dipole-dipole forces.

The Multipole Expansion

A quick review of the Multipole Expansion, with a few example calculations.

Hybrid Methods: Combining Separation of Variables and Coulomb Integrals

Coulomb integrals are hard to evaluate unless everything is very symmetric. However, we can combine them with what we've learned about separation of variables to produce a complete description of the potential in situations where we may not be able to fully evaluate the Coulomb integral itself. These notes explain the technique and give a few examples, including an (infinite Legendre series) expression for the potential due to a charged disk.

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.

Discontinuity in the Electric Field due to a Surface Change

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.

Using Taylor Series to study our results

Coulomb integrals can be hard, so we should always do what we can to make sure results we obtain this way are consistent with things we already know. This Mathematica notebook takes the charged disk example from class and uses the Series command to study its potential far away from or very close to the disk.

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 variable, trig substitution, and the method of partial fractions.

Calculating the Electrostatic Potential

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.

Using Gauss's Law

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.

Dirac Delta Examples

Here are a few examples of integrals containing three-dimensional Dirac deltas, that you can work through if you'd like a little more review.

Another Integral from Homework 3

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.

In class we worked out the electric field at a point above or below the center of a disk. These notes go through that calculation in detail, showing all the steps of setting up and evaluating the integral.

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.

The Helmholtz Theory of Vectors

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.

Examples of Line, Surface, and Volume Integrals

A quick 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.

Orthogonal Coordinate Systems

Vector Calculus

A review of orthogonal coordinate systems and vector calculus from my Phys 301 (Math Methods) course.

This is a 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 embedded 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.