electrodynamics

Assignment 12
The Poynting Vector, and Electromagnetic Waves
Due on Friday, December 8.

The last homework covers a few topics from chapters 8 (the Poynting vector) and 9 (electromagnetic waves). Solutions will be available after you turn it in on Friday, for review before the final exam.

Assignment 11
The Vector Potential, Magnetization
Due on Wednesday, November 29

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 13

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 6

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

Assignment 8
The Multipole Expansion
Due on Monday, October 30

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 23
(Problem 5 has been updated)

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 16

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, October 2

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 25

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 18
(Typo corrected on problem 2)

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 6

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 30

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.

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.

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.

Fields for Moving Point Charges
Ever wonder what the \(\vec{E}\) and \(\vec{B}\) fields produced by a moving charge look like? These notes explain scalar and vector potentials in electrodynamics, then solve Maxwell's equations for the Lorenz gauge potentials due to 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:

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).

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.

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.

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.

An Electric Field Example

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.

Coulomb Integrals

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 Few Useful Integrals

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

The Dirac Delta

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 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.

Orthogonal Coordinate Systems
Vector Calculus

A review of orthogonal coordinate systems and vector calculus for students who did not take my Phys 301 (Math Methods) course.

Line Integrals

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 various plots. Let me know if you find any typos or mistakes and I will post a corrected version.