Wangsness Homework Online

"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} \\ \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.


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 2017 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.

1August 29, 311
2September 7, 71, 2
3September 12, 142
4September 19, 212
5September 26, 282, 3
7October 3, 53
8October 10, 12Fall Break, 3
9October 17, 193, 4
10October 24, 264
11October 31, November 24, 5
12November 7, 95
13November 14, 165, 6
14November 21, 236, Thanksgiving Break
15November 28, 307
15December 5, 79

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.


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 Vector Potential, Magnetization
Due on Friday, December 1.

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

Assignment 11
Magnetostatics and Ampere's Law
Optional for Tuesday, November 21

You just completed an exam and are getting ready to leave for the Thanksgiving holiday, so handing in this assignment is optional. The material is important, thought, so you should still complete the problems even if you don't plan to hand them in.

Assignment 10
Magnetostatics and the Biot-Savart Law
Due on Friday, November 10

This short assignment is a chance to practice calculating the magnetic field using the Biot-Savart law. Similar problems may appear on our second exam, on November 15.

Assignment 9
Electric Fields in Matter
Due on Friday, November 3

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

Assignment 8
The Multipole Expansion
Due on Friday, October 27

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 October 20

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

Assignment 6
Method of Images
Due on October 12

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

Assignment 5
Electrostatic Potential Energy
Due on September 28

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

Assignment 4
Electrostatic Potential
Due on September 21

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
Due on September 14

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 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 August 29

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 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 (dates TBA) count 15% each. A cumulative final on Friday, December 16 (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.


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

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. I will also place 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.


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.

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.

The Dirac Delta

We didn't say much about the Dirac delta in class on 8/31/17, so I have written out some notes that you might find helpful when working 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.

Scholarship Review, Spring 2014

This is an edited version of a review of my scholarship, as of Spring 2014.


I started working at Drake University in August 2009. Coming out of my postdoctoral position at Iowa State University in May 2009, I was active in the combinatorial matrix theory community. With the UTMOST grant award in 2010, a $525,000 4-year NSF grant to promote open-source software and open-source textbooks, I have shifted a lot my attention recently to open-source mathematics and scientific software, such as Sage.

Here is my Curriculum Vitæ. I also have a MathSciNet publication list and an preprint list that cover most of my publications.


Combinatorial Matrix theory

In reverse chronological order, here are my papers in combinatorial matrix theory. As a general rule, I prefer to submit to open-access journals.

Steve Butler, Jason Grout, and Tracy Hall, Using variants of zero forcing to bound the inertia set of a graph, 12 pages. Submitted. Preprint
Luz M. DeAlba, Jason Grout, In-Jae Kim, Steve Kirkland, Judith J. McDonald, and Amy Yielding, Minimum rank of powers of trees, Electron. J. Linear Algebra 23 (2012), 151–163 Paper
Steve Butler and Jason Grout, A construction of cospectral graphs for the normalized Laplacian, Electron. J. Combin. 18 (2011), no. 1, Research Paper 231, 20 Paper
Jason Grout, The minimum rank problem over finite fields, Electron. J. Linear Algebra 20 (2010), 691–716 Paper
Laura DeLoss, Jason Grout, Leslie Hogben, Tracy McKay, Jason Smith, and Geoff Tims, Techniques for determining the minimum rank of a small graph, Linear Algebra Appl. 432 (2010), no. 11, 2995–3001 Preprint
IMA-ISU research group on minimum rank, Minimum rank of skew-symmetric matrices described by a graph, IMA-ISU research group members: Mary Allison, Elizabeth Bodine, Luz Maria DeAlba, Joyati Debnath, Laura DeLoss, Colin Garnett, Jason Grout, Leslie Hogben, Bokhee Im, Hana Kim, Reshmi Nair, Olga Pryporova, Kendrick Savage, Bryan Shader and Amy Wangsness Wehe, Linear Algebra Appl. 432 (2010), no. 10, 2457–2472. Preprint
Started working at Drake in August 2009; the following three papers were published before starting at Drake
Luz M. DeAlba, Jason Grout, Leslie Hogben, Rana Mikkelson, and Kaela Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph, Electron. J. Linear Algebra 18 (2009), 403–419 Paper
Wayne Barrett, Jason Grout, and Raphael Loewy, The minimum rank problem over the finite field of order 2: minimum rank 3, Linear Algebra Appl. 430 (2009), no. 4, 890–923 Preprint
D. Cvetkovic and J. Grout, Graphs with extremal energy should have a small number of distinct eigenvalues, Bull. Cl. Sci. Math. Nat. Sci. Math. 32 (2007), 43–57 Preprint

Linear Algebra Education

Other Areas

Software Code

Many of the links below lead to Github, an online repository for open-source software. My username on Github is jasongrout, which might be helpful if you are clicking on a link below and see lots of usernames.

Reviewed contributions

I have been a very active participant in the open-source mathematical software world, particularly with Sage, for over 7 years. Since I started at Drake (August 2009), I have contributed to code reviews and contributions 2,837 times on 834 separate “issues” across 43 areas of Sage, including graph theory, linear algebra, graphics, online interfaces, scientific computing packages, and many more. I also have contributed 160 peer-reviewed code contributions myself (involving adding/deleting/changing over 55,000 lines of code and documentation). This puts me in the top 15 contributors to Sage over the last 5 years out of around 700 total contributors.

In addition to working on the core Sage library, I have also worked on the following peer-reviewed projects over the last 5 years:

  • Sage Cell Server, another online interface to Sage allowing anyone to easily embed live Sage computations into any webpage: 1201 code contributions adding/deleting/changing 137,441 lines. I am the lead developer and mentored 6 undergraduate Drake students who wrote and reviewed code. I also maintain the public sage cell server at, which serves around 2,000 computations each day requested from all over the world. The cell server is used in a number of online resources, including textbooks, notes, online homework systems, and more. (Edit: As of May 2014, Andrey Novoseltsev maintains and leads development for the Sage Cell Server)
  • IPython, a very popular and easy-to-use interface to Python: 59 code contributions adding/deleting/changing 3906 lines. I helped with major enhancements like their new interactive widget framework and also contributed bugfixes and smaller enhancements.
  • Minimum Rank Library, a library for calculating the minimum rank, zero forcing numbers, and other related parameters on graphs: 58 code contributions adding/deleting/changing 4715 lines. I was the lead developer and also mentored several graduate students from Iowa State University who developed part of the library. This library is used by active researchers in exploring minimum rank problems.

Additionally, I contributed some peer-reviewed bugfixes and enhancements to the three.js 3d web graphics project and the online homework system Webwork.

Unreviewed software

  • PyThreeJS, a wrapper around three.js to provide interactive 3d graphics for the online IPython notebook: 102 code contributions adding/deleting/changing 9452 lines. I am the lead developer and mentored 3 Drake students in this project. We have just completed the initial version, and will start promoting it for wider use as a way to do 3d graphics in the IPython notebook.
  • Multi-mechanize, a suite of programs to test scalability of websites: 30 commits involving adding/deleting/changing 1191 lines.


Funded Grants

  • Rob Beezer, Jason Grout, Marja-Liisa Hassi, Tom Judson, Kiran Kedlaya, and William Stein, UTMOST: Undergraduate Teaching in Mathematics with Open Software and Textbooks, $525,000, National Science Foundation, 2010-2014. Original Proposal, Supplement. NSF Department of Undergraduate Education CCLI type 2 grant for integrating and promoting Sage and open textbooks in undergraduate mathematics curriculum. My responsibilities include implementing improvements to Sage, directing work by students, training faculty and supervising test sites, helping organize workshops, and contributing curricular materials. See also the UTMOST website.
  • Rob Beezer, Karl-Dieter Crisman, and Jason Grout, Sage: Using Open-Source Mathematics Software with Undergraduates, \$10,400, Mathematical Association of America Professional Enhancement Program, 2010. Proposal for 2010 (Supplement), Report for 2010. We introduced several dozen faculty members to using Sage in the classroom and helped them prepare curricular materials through an online workshop spanning several days throughout the summer and follow-up during the semester.
  • Karl-Dieter Crisman and Jason Grout, Sage: Using Open-Source Mathematics Software with Undergraduates, \$8,500, Mathematical Association of America Professional Enhancement Program, 2011. Proposal for 2011, Report for 2011. We introduced several dozen faculty members to using Sage in the classroom and helped them prepare curricular materials through an online workshop spanning several days in the summer and follow-up during the semester.
  • Jason Grout, Scalable Internet Interface for Sage, \$2,340, Drake University, 2011. Proposal. This grant funded two Drake student assistants to help design and write a scalable public Internet interface to the Sage, allowing Sage to embed in any webpage.

Submitted Grants

  • Rob Beezer, Jason Grout, Tom Judson, Kiran Kedlaya, Susan Lynds, Kent Morrison, and William Stein, UTMOST: Undergraduate Teaching in Mathematics with Open Software and Textbooks, $2,352,968, National Science Foundation, 2014-2019. Proposal. NSF Department of Undergraduate Education IUSE grant for integrating and promoting Sage and open textbooks in undergraduate mathematics curriculum. My responsibilities would include implementing improvements to Sage, leading development of the Sage Cell Server, directing work by students, training faculty, and helping organize workshops.


I was awarded the 2012 Spies Prize, an annual cash award to recognize “major and inspiring contributions to the development of the Sage Mathematical Software System.”

Presentations and Workshops

I've labeled presentations with one of the following areas:

  • research: pure math research, such as combinatorial matrix theory
  • teaching: improving teaching, such as explaining ideas or helping teachers know how to use electronic resources like Sage
  • technical: explaining technical infrastructure, such as how the backend of a Sage service works or the technical details of how we implemented the Sage Cell Server

I have also listed when I was a funded participant of a workshop, i.e., the workshop paid for my travel and/or local expenses.

    • Contributed teaching talk: “Online Interactive Worksheets with Sage”, MAA Mathfest, Portland, Aug 2009
    • Invited teaching talk and funded participant: “Sage and Multivariable Calculus”, Sage Education Day 1, Clay Mathematics Institute, Boston, Dec 2009
    • Invited research talk and funded participant: “Computing bounds for minimum rank with Sage”, Banff International Research Center, Banff, Canada, Jan 2010
    • Invited teaching talk: “SageTeX”, BYU faculty teaching seminar, Mar 2010
    • Invited research talk: “Computation of Minimum Rank”, AMS Central Section Meeting, April 2010
    • Contributed teaching talk: “SageTeX”, MAA Iowa Section Meeting, Oct 2010
    • Contributed teaching talk: “Eigenvalues first? Teaching linear algebra with computation, then application, then theory”, and poster for UTMOST grant in CCLI grant poster session, AMS/MAA Joint Meetings, New Orleans, LA, Jan 2011
    • Invited research talk and funded participant: “Single-cell Notebook Server”, Sage Days 31, Seattle, WA, June 2011
    • Invited teaching talk and funded participant: “Numerical Analysis in Sage”, Sage Education Days 3, Seattle, WA, June 2011
    • Organizer for Sage exhibit booth; contributed teaching talk: “Sage: free open-source math software”, MAA Mathfest, Lexington, KY, August 2011. I also gave several presentations from the Sage booth on using Sage in different classes.
    • Contributed teaching talk: “Free Online Homework with Webwork”, MAA Iowa Sectional Meeting, Pella, IA, October 2011
    • Invited research talk: “Computing inertia sets of graphs using variations of zero forcing”, AMS/MAA Joint Meetings, Boston, MA, Jan 2012. I also helped present a poster on UTMOST at the NSF poster session.
    • Co-organizer and funded participant: Sage Days 35.5, Wenham, MA, Jan 2012
    • Invited teaching talk: “Free open-source software and textbooks for mathematics”, Drake DUSCI Colloquium Series, April 2012
    • Invited technical talk and funded participant: “State of the Notebook”, Sage Days 41, Seattle, WA, June 2012
    • Invited teaching talk and funded participant: “Sage Single Cell Server”, Sage Education Days 4, Seattle, WA, June 2012
    • Co-organizer, Contributed teaching workshop: “An introduction to Sage” (with Theron Hitchman), MAA Iowa Section Meeting, Indianola, IA, Oct 2012
    • Contributed teaching talk: “Lights Out!”, Drake Math Club talk, Oct 2012
    • Invited teaching talk: “The Sage Cell Server: embedding live computations in web pages”, AMS/MAA Joint Meetings, San Diego, CA, Jan 2013. I also helped present a poster on UTMOST at the NSF poster session.
    • Invited research talk: “Bounds on maximum nullity”, AMS Central Section, Ames, IA, April 2013
    • Invited teaching talk, “The Sage Cell server and other technology in teaching”, International Linear Algebra Society, Providence, RI, June 2013
    • Invited technical talk and funded participant, “State of the Sage Cell Server”, Sage Days 48, Seattle, WA, June 2013
    • Co-organizer, invited teaching talks, and funded participant: “Sage Cell Server” and “Introduction to Interacts”, Sage Education Days 5, Seattle, WA, June 2013
    • Invited teaching talk, “Sage mathematics software in the classroom”, AMS/MAA Joint Math Meetings, Baltimore, MD, Jan 2014
  • Other conferences/workshops I participated in:
    • Participant: NSF day, U of Iowa, Oct 2009
    • Participant: AMS/MAA Joint Meetings, San Francisco, Jan 2010
    • Invited funded participant: Sage Bug Days 2, Seattle, Jan 2010
    • Participant: AMS/MAA Joint Meetings, San Francisco, Jan 2010
    • Invited funded participant: Sage Days 19, Seattle, Jan 2010
    • Invited funded participant: Math in the City workshop, Dec 2010
    • Invited funded participant: CCR workshop, San Diego, CA, Feb 2011
    • Invited funded participant: Sage Days 27, Seattle, WA, Jan 2011
    • Invited funded participant: Sage Days 29, Seattle, WA, Mar 2011
    • Invited funded participant, Sage Days 46, Hawaii, Feb 2013

Major curricular materials

  • Multivariable Calculus: Ben Woodruff and I have been working on a semester-long set of problems to explore multivariable calculus, and both of us have used versions of these notes for many years. A PDF version is gradually being superseded by an online version. These problems are based on an older lecture-style set of notes by Ben and myself. Ben wrote the initial versions of both the lecture-style notes and the problem set, and then I extensively modified and edited them.
  • Numerical Linear Algebra: I developed a set of problems to guide an undergraduate student through a semester course using Trefethen and Bau's Numerical Linear Algebra textbook. We used these notes in a problem-oriented course.


In June 2014, I went on academic leave of absence and worked with the quantitative finance research group at Bloomberg L.P. in New York to build software useful in quantitative finance. At the conclusion of the 2014-2015 academic year, I left Drake University to continue working at Bloomberg.

Citation Available
Robert Beezer, Robert Bradshaw, Jason Grout, and William Stein, Sage, Handbook of linear algebra, Second edition (Leslie Hogben, ed.), Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, (2013), 26 pages. Chapter, html version
Jason Grout, The Sage Mathematical Software System, International Linear Algebra Society Bulletin: IMAGE (Fall 2013), pp. 31–33. Article (Full bulletin)
Citation Available
David Holcomb, Eric D. Manley, Jason Grout, and Alex Hoyer, On the Integral Coding Advantage in Unit Combination Networks, 13 pages. In preparation. Preprint
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