The Physics Education Research group at the University of Colorado Boulder (PER@C) has developed and compiled a number of resources for research-based instruction throughout the undergraduate curriculum. This site includes materials developed by PER@C members as part of the Science Education Initiative and other research-based efforts, as well as materials developed by other faculty at CU Boulder.
On this site, you will find a number of materials we have borrowed or developed. Feel free to use what you like - we would like to share our materials, but also believe in giving credit to sources whenever possible (and ask that you do not use our materials for commercial purposes). We ask for your cooperation in not making any solutions you may create for the homework (and exam problems, clicker questions, etc…) available on the open web, out of respect for instructors and students at other institutions, and for maintaining the integrity of our research.
If you have questions, contributions, bug-catches, etc, please contact steven.pollock (at) colorado.edu Thanks!
Modern Physics is the third semester in our three-semester sequence of introductory physics courses. Materials include the following topics:
special relativity, photoelectric effect, spectra, lasers, Bohr and deBroglie models, Stern-Gerlach, entanglement and single-quanta experiments, matter waves and the Schrödinger equation, tunneling (α-decay, STM's), hydrogen atoms and molecular bonding, conductivity, semiconductors and BECs.
Materials were originally developed for a course for engineering majors, with a focus on applications, in 2005 and 2006 by Wieman, Perkins, and McKagan (McKagan et al. 2007). They were later adapted for a course for physics majors to include special relativity by Finkelstein, Bohn, and later Rogers, Schibli, and Dessau. Finkelstein and Baily made further research-based adaptations to include content on quantum interpretation (Baily and Finkelstein 2010). Later updates by Finkelstein added a unit on diversity.
On this page, you will find a number of materials we have borrowed or developed. Feel free to use what you like - we would like to share materials, but also believe in giving credit to sources whenever possible. We ask for your cooperation in not making any solutions you may create for the homework (and exam problems, clicker questions, etc…) available on the open web, out of respect for instructors and students at other institutions, and for maintaining the integrity of our research.
The first set of materials is older, from the Science Education Initiative team (including C. Wieman, K. Perkins, S. McKagan, N. Finkelstein, and many others). It includes materials from multiple instutions dating back to 2009, and includes some homework and exam materials, lecture notes and slides, and more.
The second set of materials is from a recent implementation of the course at CU Boulder by Noah Finkelstein.
Quantum Mechanics Conceptual Survey (QMCS) assesses conceptual material typically taught in modern physics. Topics include wave functions, probability, wave-particle duality, uncertainty principle, infinite square well, one-dimensional tunneling, energy levels. You can learn more and download the assessment from PhysPort using the link above.
The archived modern physics materials are available together as a package in the Materials tab. The course calendar below (for SP11 - with SR) is the simplest way to browse the most current materials - the Materials download (above) contains the files.
Written homework was assigned on Tuesdays, and due at the beginning of class on the Thursday in the following week (i.e., 10 days later). Students were expected to complete the readings before the lecture.
WEEK |
Tuesday Lecture | Thursday Lecture | Homework |
1 |
(No class) | 1. Introduction, math review |
HW 1 |
2 |
2. E&M review, waves & wave equations |
3. Interference, polarization, |
HW 2 |
3 |
4. Michelson-Morley experiment, SR postulates |
5. Time dilation, length contraction |
HW 3 |
4 |
6. Lorentz transformations, spacetime, addition of velocities |
7. Relativistic momentum, energy |
No HW |
5 |
8. Modeling in physics, intro to quantum |
Exam 1 |
HW 4 |
6 |
9. Photoelectric effect 1 |
10. Photoelectric effect 2, photons |
HW 5 |
7 |
11. Photons, atomic spectra |
12. Lasers |
HW 6 |
8 |
13. Bohr model, de Broglie waves |
14. Stern-Gerlach experiments |
HW 7 + |
9 |
15. Repeated spin measurements, probability |
16. Entanglement, EPR, quantum cryptography |
HW 8 + |
10 |
17. Single-photon experiments, complementarity |
18. Electron diffraction, matter waves, |
HW 9 + |
11 |
19. Matter waves, Review for Exam 2 |
Exam 2 |
No HW |
12 |
20. Wave equations, Schrodinger equation, |
21. Infinite/finite square well, tunneling |
HW 10 |
13 |
22. Tunneling, alpha-decay |
23. Radioactivity, STM's |
HW 11 |
14 |
24. Hydrogen atom 1 |
25. Hydrogen atom 2 |
HW 12 |
15 |
26. Multi-electron atoms, periodic table, |
Exam 3 |
No HW |
16 |
27. Molecular bonding, Bose-Einstein Condensates |
Review |
(Final Exam) |
The online simulations listed below were used in both lectures and homeworks for the Spring 2011 modern physics course at CSM. Most of these links lead directly to the PhET Interactive Simulations project, online simulations developed at the University of Colorado, many of them specifically for this course. There are a number of sims for all kinds of quantum phenomena that we didn't use, as well as general physics content. We would also recommend the simulations from the The Quantum Mechanics Visualisation Project at the University of St Andrews.
Note that some older sims are java or flash. (Your mileage may vary running those on modern browsers - if you want to use these in classes be aware that e.g. some will not work on phones or pads)
Weeks 1-2: (Pre-Quantum)
Special Relativity:
We're aware of only a few simulations for topics from special relativity [please let us know about ones you think are good]. The special relativity lecture slides (CSM SP11) are nicely animated, but not particularly interactive. We have not tested them with students, but there are some interesting visualizations at:
Week 6:
Week 7:
Week 8:
Week 10:
Week 13:
Week 14:
(This is an early draft from 2006)
Selected topical learning goals for Phys 2130
1. Wave function and probability
2. Wave-particle duality
3. Schrodinger Equation
4. Quantization of energy/quantum numbers/unique states
5.Uncertainty principle
6. Superposition
7. Operators and observables
8. Measurement
The papers linked below explain much of the process and rationale behind the transformations. The first part (2006) reports on the state of affairs following the first year of the process, after the course had been taught twice (in the FA05 and SP06 semesters). The second part (2011) details additional changes that were made to the materials as part of Charles Baily's dissertation project on quantum perspectives.
At CU Boulder, sophomore Classical Mechanics spans two semesters, explicitly adding coverage of a number of math tools that will be (re)encountered throughout the remainder of a typical physics major's career, introducing them in the context of Classical Mechanics.
Classical Mechanics/Math Methods 1 includes Newton's laws with velocity dependent forces, rockets, energy and gravity, and oscillations with damping and drivers. Classical Mechanics/Math Methods 2 continues with Lagrangian and Hamiltonian formalism, rigid body rotation, normal modes, orbits, non-inertial frames, linear algebra and matrix methods, and calculus of variations.
The Classical Mechanics 1 course was developed through the Science Education Initiative starting in 2009 with work from S. Pollock, S. Chasteen, R. Pepper, A. Marino, D. Caballero and many others.
The course was updated by S. Pollock and E. Neil with additional in-class tutorials and lecture notes in 2022. The Classical Mechanics 2 course was developed by several faculty at CU who shared their materials, especially E. Neil, and is not associated with any education research projects.
For details about either semester, click on the appropriate course link above.
E&M at CU is a two-semester sequence of junior-level classical electricity and magnetism.
Content coverage follows the textbook of Griffiths, Introduction to Electrodynamics)
E&M 1 covers electro- and magnetostatics, roughly Ch 1-6 of Griffiths.
E&M 2 covers electrodynamics, roughly Ch 7-12 of Griffiths.
For details about either semester, click on the appropriate course link above.
Reformed course materials were developed through the Science Education Initiative starting in 2007 with work from S. Pollock, S. Chasteen, M. Dubson, C. Baily, X. Ryan and many others.
Quantum Mechanics 1 is the first semester of our two-semester sequence of quantum mechanics.
This tab contains links to materials for a variety of (mostly) undergraduate courses offered at CU Boulder.
Materials in this page are not "research-validated", they did not arise from the Science Education Initiative. They are a collection of informal materials that might prove useful if you are teaching a student-centered large University-level course.
Use what you like - give credit to sources when feasible. We ask that you do not use our materials for commercial purposes. We also ask for your cooperation in not making any solutions you may create for the homework (and exam problems, clicker questions, etc…) available on the open web, out of respect for instructors and students at other institutions, and for maintaining the integrity of our research.
Our material sets include concept tests and lecture notes, and sometimes more (e.g. course goals, etc) Courses are identified by topic, with details in each tab.
We teach a variety of introductory courses at CU. The calculus-based sequence (Phys 1110, largely engineers) serves over 1000 students/semester, split into sections of 300 (3x 50 minutes/week), and recitations (1x 50 minutes of UW Tutorials) of 28 students. A very similar course for our majors serves (Phys 1115) about 125 students/year, same format.
The main Physics 1 download is from a recent implementation of the major's course.
Materials should still be useful for any calculus-based course (and with modification, an algebra-based course, as we don't require Calculus as a prerequisite, so we use minimal amounts of it)
These materials are research-informed but not research-validated. Primary contributors are S. Pollock, M. Dubson, and D. Bolton, with contributions from many others.
Physics 2 is Electricity and Magnetism.
We teach a variety of introductory E&M courses courses at CU. The calculus-based sequence (Phys 1120, largely for engineers) serves just under 1000 students/semester, split into sections of ~300 (3x 50 minutes/week), and recitations (1x 50 minutes of UW Tutorials) of 28 students. A very similar course (Phys 1125) for our majors serves about 125 students/year, same format.
We have an (old) collection of materials from an implementation in 2007, working on some updates.
Still coming - please check back (or contact steven.pollock (at) colorado.edu if you are in a hurry!)
These materials are research-informed but not research-validated. Primary contributors are S. Pollock, M. Dubson, and D. Bolton, with contributions from many others.
Materials from a graduate level course at CUB on Physics Education Research (cross-listed for advance undergraduates) Designed by Noah Finkelstein.
Still coming - please check back (or contact steven.pollock (at) colorado.edu if you are in a hurry!)
We teach an interactive large-lecture course called "Light and Color" to about 100 students/term, mostly non-science majors. The course is an introduction to the science of optics, with no prerequisites (and very light on math or formalism). It uses a variety of readings, including from a (free) OpenStax textbook (College Physics 2e).
Downloads below are from a recent implementation taught by B. Wilcox.
The materials are (largely) not research-validated, but were inspired in part by earlier course transformations at CU Boulder from 2010-2017 taught by Stephanie Chasteen, CharlesRogers, Katie Hinko, and Cindy Regal, with contributions from other instructors.
This course is a large introductory level class aimed at non-science majors. The materials shared here are not research-based in any way, just some materials from when S. Pollock taught the course in 2007.
We teach a senior-level thermodynamics and statistical mechanics course for physics majors, following the textbook "Introduction to Thermal Physics," by Daniel Schroeder.
Materials are not research-validated, merely shared by faculty at CU including Michael Dubson, M. Hermele, V. Gurarie, and most recently Bethany Wilcox (whose latest version is featured in the download) The course has interactive elements (clicker questions, and in-class Tutorials) developed by various faculty.
This is the second semester in our two-semester sequence of junior-level classical electromagnetism courses for physics majors, electrodynamics. Content coverage follows Ch 7-12 of Griffiths, Introduction to Electrodynamics). This includes time-dependence in Maxwell's equations, AC circuits, conservation laws, EM waves in vacuum and media, potentials and gauge transformations, radiation, and special relativity.
On this page, you will find a number of materials we have borrowed or developed. Feel free to use what you like - we would like to share materials, but also believe in giving credit to sources whenever possible. We ask for your cooperation in not making any solutions you may create for the homework (and exam problems, clicker questions, etc…) available on the open web, out of respect for instructors and students at other institutions, and for maintaining the integrity of our research.
See C. Baily, M. Dubson and S. Pollock, "Research-Based Course Materials and Assessments for Upper-Division Electrodynamics (E&M 2)" PERC proceedings 2012 for more early development information.
Colorado UppeR-division ElectrodyNamics Test (CURrENT) assesses basic topics from junior-level electrodynamics, where the focus is on gauging student understanding of fundamental concepts, and whether they can complete basic advanced E&M tasks. The topic coverage in this assessment is limited to core material that is likely to be taught at most institutions, since we’ve found there to be some variance due to institutional peculiarities. There is also a short pre-test on a more limited set of questions. You can learn more and download the the open-ended version (with scoring rubrics) from PhysPort using the link above.
Online, multiple-response versions of both pre- and post-tests are under development, with a beta version implemented on Qualtrics. Contact Steven.Pollock (at) Colorado.edu if you would like to try this in your class.
E&M 2 is the second semester of our junior-level classical electromagnetism sequence. We have compiled a number of resources which can be easily incorporated into a standard university class structure. These resources were designed to encourage students to be active participants in the learning process. See the Materials tab for all materials in one download.
Here, we briefly summarize our transformations with tips to implement them yourself.
The primary text we used for this course is D.J. Griffith, Introduction to Electrodynamics (Prentice-Hall, Upper-Saddle River NJ, 1999) [Ch. 7-12]).
The following additional textbooks were recommended by electrodynamics instructors at CU Boulder, and various physics faculty at outside institutions:
The bulk of the material in this course is fairly canonical across universities. At the University of Colorado, the content coverage and order closely follows chapters 7-12 of Griffith's text.
We begin with an introduction to electrodynamics (EMF, Faraday, Maxwell's equations with time dependence), add a practical unit on inductors in circuits, then move to conservation laws, Poynting vector and the stress tensor, then EM waves, time-dependent potentials and fields, radiation, and end with special relativity.
Additional commentary on the presentation of certain topics can be found in the Materials download tab.
Our course requires that students have passed E&M 1 (electro- and magnetostatics) before starting the second semester. Mathematical expectations are no different than for E&M: 3 semesters of calculus, plus differential equations. Faculty may choose give a content pretest to students to both (a) assess where students are weak, and (b) send students the message that this is material they should already be familiar with.
There is a general consensus among faculty that the bulk of the learning in this course comes from doing the homework. This course is where students learn a certain level of sophistication in solving problems (see Learning Goals tab). Homework should reflect that higher expectation. We have compiled a homework bank of useful problems designed to target these higher level goals. Additional ideas for creating homework sets can be found in the Course User's Guide. Some were written by CU faculty, but many were also borrowed and adapted from others (including textbooks), modified to emphasize sense making and/or connections to real-world situations.
We ask for your cooperation in not making solutions to these homework questions available on the open web.
These were given at the very beginning (as in, the first day), turned in on the second day of class. In the end, both ways of doing it provided a great deal of information about what students were capable of after a long break between semesters, and helped us decide how to approach the first week of the semester, which was devoted to review topics from electrostatics.
See the Materials download, where our homeworks are available in Word and pdf versions from several semesters. (Feel free to adapt, alter, and improve. Do try to significantly change our wording, and do not post CU solutions, thanks.)
Preflights are short questions or tasks asked online, used to orient students to upcoming material. They are meant to encourage reading the textbook before coming to class, and to reflect on key points in the material. Responses can be used by an instructor to inform their preparations for upcoming lectures, by focusing on specific difficulties students have with new material.
These preflights were loosely modeled on ones from the US Air Force Academy, an upper-division offshoot of Just-in-time Teaching (JITT). For more information, see: G.M. Novak, E.T. Patterson, A.D. Gavrin and W. Christian, Just-in-time Teaching (Prentice-Hall, Englewood Cliffs NJ, 1999), and other publications associated with these authors.
Our preflight assignments were due several hours before class early in the week. They represented a small percentage of their total grade, and were graded strictly pass/fail, based effort and not correctness. Preflights at CU were administered online, but the Materials download contains screenshots or pdf versions.
Comment: Preflights were not particularly popular with our students, though they were extremely useful to us as instructors and education researchers. In other words, despite the valuable information we received from students, they did not claim to find the preflights particularly helpful for their learning (only 14% of SP12 and 30% of FA11 students rated preflights as either useful or very useful for their learning). This may in part be due to low student participation (they needed constant reminders in both courses that the preflights were due each week), and many students reported being averse to doing work over the weekend. In later courses we have found that discussing student responses in class, responding individually to some every week, and promoting their value go a long way in improving student attitudes.
In lieu of “office hours” in a faculty office, our courses hold twice-weekly homework study (or "help") sessions outside of class. We devote 1-2 hours each session, timed to be convenient to a large fraction of the class (based on polling), the 2 days before homeworks are due. Learning assistants or TAs can staff these sessions if you have them available. Our sessions were held in a large room with sufficient table space for students to work together on the homework problems, with occasional guidance from an instructor. This was particularly enlightening to us as instructors and researchers, since we could observe students as they were engaged with the work. They were also extremely popular with students (84% of students who came rated these sessions as useful or very useful for their learning.) Attendance was typically ~1/3 of the class, we found it important to remind students and strongly encourage them to come.
There are a variety of lecture techniques that have been shown to be useful in student engagement.
Clickers are wireless personal response systems that can be used in a classroom to anonymously and rapidly collect an answer to a question (usually multiple-choice) from every student. This allows rapid reliable feedback to both the instructor and the students. Alternatively, clicker questions can still be used without the personal response system by using colored cards or hand signals. See the Colorado Science Education Initiative website for additional information and resources for effective use of clicker questions.
Many consider Richard Hake’s 1998 AJP paper a wake-up call for physics instructors, demonstrating that traditional teaching methods (lectures with occasional demonstrations) are largely ineffective in fostering a deep understanding of physics. Hake analyzed introductory physics courses across institutions (See Ref 1 below), categorizing them as traditional or incorporating active engagement techniques. Students completed the Force Concept Inventory (FCI) pre- and post-instruction, testing mastery of Newton’s laws. Results were measured by learning gain <g>, which compares how much students learned to how much they had yet to learn at the start. This relative measure accounts for variations in students’ prior knowledge, enabling cross-institutional comparisons of instructional effectiveness.
Traditional lecture courses showed consistently low learning gains, typically in the 20–30% range—indicating students learned only about one-quarter of what they didn’t already know. While some active engagement courses also had low gains, the majority achieved two to three times higher learning gains than traditional ones. These findings have been replicated in other studies (Ref 2 and 3 below), which also show that interactive tutorials improve long-term retention of conceptual understanding in E&M compared to traditional recitations. These results highlight that while active engagement is critical for learning, how it is implemented matters.
Active engagement is rarely applied to upper-division courses, despite evidence supporting its effectiveness at the introductory level. Pollock (Ref 4) compared pre- and post-instruction scores of junior-level electromagnetism students on the Brief Electricity and Magnetism Assessment (BEMA), a measure of fundamental concepts from introductory E&M. He found no significant improvement in scores after traditional advanced E&M instruction, suggesting that the teaching approach did not help solidify students’ grasp of foundational concepts. This raises concerns about whether students can develop a deep understanding of more advanced topics without mastering the basics.
If active engagement benefits introductory courses, why not apply it to advanced ones? Instructors often view upper-division courses as fundamentally different. The topics are more abstract and mathematically formal, and students have already demonstrated aptitude by progressing to this level. Smaller class sizes allow for more interaction, and instructors often see these courses as preparation for graduate study. Many faculty teach upper-division courses as they themselves were taught, believing the nature of the material requires a more traditional approach.
However, research shows many advanced students are not learning as effectively as we hope (Ref 5). Active engagement, such as clicker questions, provides opportunities for students to recognize gaps in understanding and apply concepts in real-time. Even in small classrooms, students often do not realize they are confused or know what questions to ask without structured prompts. Clicker questions also help instructors gauge comprehension during lectures. Active engagement fosters scientific argumentation—an essential skill developed through practice as students articulate and defend their understanding.
Student response to active engagement in upper-division courses has been overwhelmingly positive at CUB. Surveys at CU found that 80% of students who experienced clicker questions in advanced courses felt they enhanced their learning compared to lectures alone, with around 15% neutral. Few believed clickers detracted from their learning experience. This feedback underscores the potential of active engagement to improve advanced instruction, building on its proven success at the introductory level.
Just as interesting is the fact that, as with many instructors, students who had never used clickers in an upper-division course were skeptical about their usefulness in that context. Students were asked following a popular pure-lecture course on quantum mechanics if they would recommend using clickers in advanced courses, with the majority responding either neutrally or not recommending their use.
In contrast, most students from a quantum mechanics course that had used clickers during lecture would recommend using them in advanced courses. Students who had used clickers in a first-semester electromagnetism course continued to favor their use after taking the second-semester course (E&M II), where they were not used.
Having motivated the use of clicker questions in upper-division courses, a natural question is: how can they be effectively incorporated into lectures? Active engagement doesn’t require instructors to give up lecturing entirely, and clicker questions can be judiciously inserted into a lecture at key points for a number of purposes, for example: to check for conceptual understanding, to have students apply a concept to a new situation, to underscore an important idea in a long derivation, or to help them make connections between a physical system and the mathematics used to describe it. Specific examples of these are given below.
Clicker questions are often only associated with testing conceptual understanding – asking students to come up with an answer without resorting to mathematical calculations. The sample question below asks whether the electrostatic field of a charged capacitor plate is capable of generating a non-zero EMF around a closed loop. This question was used in two separate electrodynamics courses at CU, and in both cases ~25% of students got this wrong; when asked, many of them intuitively knew that a charged capacitor shouldn’t be able to drive a current around a loop, but some weren’t considering the existence of fringing fields, or couldn’t visualize how they were causing the line integral to vanish.
Students can also test their understanding of important concepts by applying newly derived results to a novel situation. In one of our classes, following a standard derivation of the magnetic field due to a long, straight wire using Ampere’s law, students were asked in the following question about the magnetic field at a point near a long wire that takes a 90 degree turn. Answering this question requires students to consider the role of symmetry in deriving the answer for the straight wire, and also the directional variation in the field produced by a section of current according to the Biot-Savart law. Because the vertical section contributes nothing to the field at point s, the field can be found by simply dividing the expression for a straight wire by half. Most students initially got this question wrong.
Sometimes there is an essential point that students must keep in mind in order to follow along with an important derivation. When discussing Maxwell’s modification to Ampere’s law, arguments involving the continuity equation required students to know that the divergence of the curl of a field is identically zero. The question below was used to underscore the importance of this point in the derivation, and to be sure that students realized this was a general result, and not specific to any particular type of field.
Clicker questions can also be used to help students make a connection between a physical situation and the mathematics used to describe it. A mathematical formula for the electric field is required to answer the question below, which essentially asks them to relate the curl of an electric field to the rate of change of magnetic flux in that region. Even though the field may appear “curly” to students, the 1/distance dependence of the field actually makes its curl everywhere zero except at the origin. This answer can be gotten without looking at the explicit expression for the curl of a field in cylindrical coordinates, by using a specific line-integral as an example, where the increase in arc length along a curve at a greater distance from the origin exactly cancels the diminished magnitude of the field. Before discussion, students in the FA11 and SP12 E&M II classes at CU were evenly split between the curl being zero (A) and non-zero (B).
Clicker questions (with notes and comments) are all available from our Materials download tab.
When solving a problem on the board, the lecturer can pause and ask the class for the next step. If the course culture has included the use of clicker questions, so that students are habituated to actually engaging with this sort of question (instead of waiting for the smartest student to answer), then this type of discussion can occur without the use of actual clickers in every instance. The class should be given a time limit (e.g., “You have 30 seconds; write down your answer”) to focus their discussion. We find that students are more likely to actually write something down on paper if the lecturer leaves the front of the room and talks briefly to students in the middle of the room.
In addition to clicker questions, faculty can pose open-ended questions (non multiple choice) for discussion in class, providing students an opportunity to engage with the concepts in class. The more that instructors are clearly open to discussion in class, the more students will feel comfortable posing spontaneous questions.
Students can read the chapter as they work on the problem set. It may be useful to encourage students to read the chapter before lecture, if the professor does not intend to reiterate material from the book in lecture. In that case, lecture may be spent in productive discussion and engagement with the material. Students can easily read derivations and similar content in the book, and so professors may decide how much of that content should be included in lecture.
We have successfully used whiteboards and student work at the blackboard in class and out of class. Large (2x3 foot) whiteboards provide a convenient public work space for group activities. Small (1x1 foot) whiteboards work well for individual or partner work while still allowing instructor to quickly see what students are getting in a lecture (by walking around to individual whiteboards or by asking students to “publish” their results by holding up their whiteboards).
Additional information on some of the advantages and disadvantages to whiteboard activities can be found in the Course User's Guide or you can browse our in-class activities for specific examples.
Tutorials are conceptually focused worksheet activities designed to be done in small groups and target known student difficulties. They are designed to be completed in a 50 min co-seminar; however, some instructors at CU have incorporated Tutorials into lecture. See the next section below for many more details (including how to run them, what topics are available, and some instructor guides with timing suggestions for each one)
In the reformed course, we encouraged students to work in small groups on the homework. They learn by peer instruction with occasional input from the instructor, as in the tutorials. Each group may have a group-sized whiteboard (see above), and the staff present do not work out problems on the board. We try to offer two homework help sessions – two nights and one night before the homework is due.
Use the Materials tab to find Tutorial source files.
Our guiding principle in creating these activities was that students would gain more from being active (instead of passive) participants in the classroom. The tasks are focused on promoting understanding of important topics from second-semester E&M, and/or guiding students through derivations that would typically be done during lecture. Many were inspired by in-class observations of student difficulties, and have been tested in focus-groups and in the classroom.
We ask for your cooperation in not making solutions to these tutorials/activities available on the open web under any circumstances - additional reasons for this (beyond the obvious) are addressed in the implementation notes (Section I, below).
A short (4-page) paper on the process of developing these tutorials: "Developing Tutorials for Advanced Physics Students: Processes and Lessons Learned", Charles Baily, Michael Dubson and Steven J. Pollock PERC Proceedings 2013 pdf
These tutorials/activities were developed for use during class, to augment or replace standard lectures on the topics they address. This style of implementation is in contrast to introductory-level Tutorial settings, being separate from the lecture portion of the class, but they could certainly be adapted for use in such environments, and we encourage instructors to do so.
Depending on the topic, they require anywhere from 10 to 50 minutes of class time, and versions of them were recently used intermittently during a course having 50-minute class periods (three times per week over a 15-week semester). ~40% of CU SP12 lectures included tutorial activities. We encourage you to personalize the materials, including shortening them considerably (Over time, we are finding that limiting Tutorials to just one page front and back makes a good length)
We orient students to the activities before they began with a concept test and/or discussion. They are implemented at appropriate times during lecture, often during the middle, sometimes at the end of class.
Incorporating these student-centered tasks into the classroom was sometimes challenging, and we describe below some lessons we have learned about getting the most out of the time spent.
(1) Sell students on group work. Students will have their own ideas about what a junior-level physics classroom should be like, and some may at first be reluctant to engage in activities that differ from the standard lecture format (even when they are familiar with them from introductory courses, and particularly if they associate them only with “lower-level” work). Aside from the belief that they will gain more through active participation, instructors may also remind students that scientific argumentation (oral or written) is a skill that is developed with practice, and that scientists work almost exclusively in group settings. Stronger students benefit from working with weaker students (and not just the other way around) since, as we should know from our own teaching experiences, they will never understand something so well as when they can explain it to someone else!
(2) Hand out just before activity begins. We’ve found that handing out the printed activities at the beginning of class (or a significant amount of time before starting them) is not ideal. There will inevitably be some students who immediately start reading through the pages or working the problems, and mostly tune out the instructor from that point on, so instructors should be aware they might not have the undivided attention of the class once the activity sheets are in front of students. This can also discourage students from collaborating with others at their table, since they’ll be ahead of everyone else and may be reluctant to go back.
(3) Keep it closed note. We have tried to provide students with sufficient information to complete these activities without having to refer to their notes. Some of the tasks do require them to recall facts from memory, but this is only in cases where we feel they should have this knowledge at their fingertips, and instructors can certainly write out necessary equations on the board if they wish. If there are instances where students feel they must refer to outside sources, this should be an indication to them that they may need to devote a little more time to studying that particular topic. We actively discouraged them from copying equations or following examples from the textbook, since this does not involve the kind of understanding we are trying to promote.
(4) Introduce the activities. Students may require some kind of orientation to the topic at hand, or need an important piece of information to get started; they may also need you to be explicit in connecting the tasks as a whole to your overall learning goals. We have tried to be as clear as possible in the problem statements, and their wording has (in most cases) already been tested with students, but what seems “obvious” to instructors may not be so for students. We have also noticed that, even at the junior-level, some students don’t always read each problem statement completely, often only skimming the words and trying to glean as much information as possible from the diagrams.
(5) Activities may take longer than anticipated. All of the activities (except where explicitly noted) have been validated through student interviews and field-tested in a classroom setting. The summary that accompanies each tutorial has an estimate of the amount of time it should take for most students to complete the entire activity, but an actual implementation may take more (or less) time than anticipated. We notice there is a tendency for instructors to underestimate the amount of time it will take students to complete these activities; a general rule is that students usually take around 10 minutes per page.
(6) Use challenge problems, or create new ones. When students are working at their own pace, there will always be groups who are much quicker than others to complete the tasks. To keep these students using their time productively, many of the activities have one or two challenge questions at the end, which usually involve taking their conclusions a step further. If there isn’t a challenge question, instructors should be prepared with a question or task for students that builds in some way off of the tasks they’ve just completed.
(7) Don’t provide written solutions. There have been studies that suggest students will learn and retain more when they are not given written solutions to tutorials, though it is essential that tutorial instructors ensure that students are arriving at correct answers as they progress through the tasks. Some students may feel frustrated by this policy, but we suspect that referring to an answer key while studying may short circuit an important aspect of the learning process, namely arriving at a correct answer through their own reasoning, and being able to justify the correctness of that answer. For our class, activities were posted on a secure site for students who were unable to attend, and we encouraged them to speak with us (and each other) outside of class about any questions they might have. Most importantly, they should ask questions during class time, when they recognize that they’re confused. We do ask that you not post solutions to these activities on the open web under any circumstances, out of respect for instructors at other institutions, and for maintaining the integrity of our research.
The ordering of topics for these activities follows the presentation in Griffiths (except for AC circuits, which is sparsely covered in his book). The activities are usually sufficiently self-contained that they can be used independent of each other, but they sometimes come in two parts, or use language we expect students to be familiar with from earlier tasks. We make note of this in instructor notes when applicable.
The general topics covered in each of the tutorials/activities are listed below, along with the estimated time it will take for most students to complete them.
00 - Review Material | |
A - Divergence & Stokes Theorems | ~ 15 min. |
B - Gauss' Law | ~ 15 min. |
C - Ampere's Law | ~ 15 min. |
01 - Current Density |
~ 15 min. |
02 - Ohm's Law |
< 50 min. |
03 - Faraday's Law |
< 30 min. |
04 - Complex Exponentials |
< 50 min. |
05 - Complex Impedance |
< 50 min. |
06 - Maxwell-Ampere Law |
|
A - Part 1 | ~ 25 min. |
B - Part 2 | ~ 25 min. |
07 - Boundary Conditions |
< 50 min. |
08 - Energy Flow in a Simple Circuit |
< 40 min. |
09 - Linear Operators |
< 15 min. |
10 - EM Wave Equation |
~ 10 min. |
11 - Complex Plane Waves |
< 40 min. |
12 - Reflection & Transmission |
|
A - Normal Incidence | < 50 min. |
B - Oblique Incidence | ~ 30 min. |
13 - Gauge Invariance |
< 50 min. |
14 - Retarded Potentials |
~ 30 min. |
15 - Special Relativity |
|
A - Length Contraction | < 15 min. |
B - Inelastic Collision | ~ 10 min. |
C - Velocity Transformation | ~ 10 min. |
These are meant to be short review activities, so the time-estimates are based on students already having a reasonable familiarity with these topics from the first semester of the course.
Topics: Divergence theorem, Stokes’ theorem, Gauss’ law, Ampere’s law.
Summary: Students are asked to state the divergence theorem and Stokes’ theorem, and then work backwards from the integral forms of Gauss’ law and Ampere’s law to derive these expressions in differential form.
Comments: Many students will have difficulty recalling these two mathematical theorems from memory, but we encourage them to do this because perpetually copying out of a book does not demonstrate understanding, and we also believe that writing them down should be straightforward if they genuinely understand what they mean. Students are typically asked to derive the integral forms from the differential forms, and these tasks have them do it in the other direction. The greatest difficulty for them was in justifying dropping the integration symbols in the final step of their derivations; students may recognize that two integrals being equal doesn’t necessarily mean the integrands are equal, yet still make the mistake of implicitly assuming this.
Topics: Gauss’ law, symmetries, electric field from a line charge distribution.
Summary: Students consider the symmetry of a line charge distribution to argue for why the electric field is entirely in the radial direction, and why a Gaussian cylinder is needed to solve for the electric field (instead of a sphere or a cube). Students are then asked to recall Gauss’ law in integral form, find the charge contained in a section of wire, and solve for the electric field.
Comments: Many students had difficulty articulating their reasoning on the symmetry questions, and were more inclined to argue in terms of the curl (or closed line integral) of an electrostatic field being zero. A significant number of students will believe that the electric field can be solved for using Gauss’ law and a non-symmetric surface, but that we don’t use such surfaces because the integral would be too difficult to calculate. All of this indicates that students may have the rote application of Gauss’ law down, without necessarily having a strong grasp of the important role of symmetry when calculating fields.
Topics: Ampere’s law, symmetries, magnetic field of a long wire.
Summary: Students first argue for why the magnetic field is entirely in the tangential direction for a straight current-carrying wire. They are then asked to recall Ampere’s law in integral form, and solve for the magnetic field around the wire.
Comments: The previous activity on Gauss’ law was more explicit about making symmetry arguments, and many students may still do this after having completed that prior activity. Others were more comfortable thinking in terms of there being no magnetic charges, and the curl (or closed line-integral) of the B-field being zero where there is no current (enclosed). Instructors should be aware that understanding the symmetry arguments in applying Gauss’ law doesn’t necessarily translate to the context of Ampere’s law. A challenge question at the end asks if Ampere’s law can be used to find the B-field at the center of a circular loop of current, which inspired a great deal of good discussion/questions.
01 – Current Density & Charge Conservation (~15 minutes)Topics: Current density, conservation of charge (continuity equation).
Summary: Students first consider a cylindrically symmetric conductor having three regions of different cross-sectional area. The task here is to rank order the three regions in terms of several physical quantities in those regions: conductivity, total current, current density and electric field. The remaining tasks connect the flux of the current density through a closed surface to the rate of change of the charge enclosed within the volume.
Comments: These tasks were overall relatively straightforward for students. A common difficulty that arose had to do with whether the outward flow of current corresponding to positive flux, and if (- dp/dt) is a positive quantity. A challenge question at the end has them convert the integral form of the continuity equation to its differential form.
Topics: Ohm’s law, continuity equation, boundary conditions on the electric field inside a conductor
Summary: A steady current flowing through a cone-shaped resistor is used as the context for addressing the implications of the microscopic version of Ohm’s Law (J = σE). The initial multiple-choice question orients students to the situation by having them consider the current density inside the resistive material. They are then led to make conclusions about the electric field and local charge density inside the resistor by using Ohm’s law in conjunction with the continuity equation and Gauss’ law. Students are presented with two possible configurations for the electric field inside the conductor, and are asked to identify which aspects of those configurations are allowed, and which are precluded by boundary conditions or conservation of charge/current. The final activity asks them to interpret a graph of the correct field and equipotential lines inside the resistor in terms of the concepts discussed in the previous sections.
Comments: Instructors should be sure that students reconcile their mathematical conclusions (Div.E = 0 inside the resistor) and the fact that the field lines are spreading outwards (which may look to them like a “diverging” field). It is not essential that the field lines drawn by students on the second page are completely correct before moving on – we just want them to develop some kind of expectation for what they ought to look like.
Topics: Faraday’s Law, fields of a solenoid with time-varying current.
Summary: Students first sketch the B-field for a long solenoid, and then consider whether there is a non-zero electric field anywhere in space when the current in the solenoid is changing with time. They then use Faraday’s law in integral form to compute the electric field inside and outside the solenoid, and sketch the induced field as a function of distance from the center.
Comments: This is a shortened version of a tutorial on EMF from a series created by the University of Colorado for the first semester of this course. The biggest conceptual difficulty for students has been with the idea that there is a non-zero electric field in a region of space where the magnetic field is zero (outside the solenoid). This can lead to good discussions on the difference between the differential and integral forms of Faraday’s law. There have been a few students who were concentrating only on the electric field driving the current in the coil, and weren’t thinking there could be an electric field anywhere but inside the wire. This can lead to interesting discussions about the relative magnitudes of the induced electric field and the field driving the current, and how this could depend on the dimensions of the solenoid or the rate of change in the current.
Topics: Complex exponentials as oscillatory functions, representations of complex numbers, simple AC circuit with resistor.
Summary: The first tasks are meant to help students gain some familiarity with complex exponentials as oscillatory solutions to differential equations. They first consider similarities and differences between exponential and trigonometric functions as solutions to a first-order equation, then similarly for the behavior of their second-derivatives. Students then perform a few basic tasks involving different representations of numbers in the complex plane, and draw conclusions about the direction of rotation over time for an arrow representing a complex exponential function. The final task applies to a simple AC circuit, where students must find the magnitude of the current through the resistor at a specific time.
Comments: The first task may seem “simple”, but we were surprised by the amount of time that some students took to find the first-derivatives of the functions given; a sign error here and there was common. Students didn’t necessarily have problems completing the exercises on complex numbers, but seem to require more practice in order to be comfortable with them; some will be rusty on the rules for multiplying exponentials functions. Questions about a vector rotating in the complex plane are given in anticipation of their use in future tutorials (#5-Complex Impedance, #7-Boundary Conditions, and #12-Reflection and Transmission). We’ve found this to be a very powerful visualization tool for students when working with oscillatory functions. Some students have shown a tendency to confuse their use of complex exponentials in quantum mechanics (multiplying by the complex conjugate to find a physical quantity) when finding the physical voltage or current represented by a complex exponential (instead of looking at just one component).
Topics: Complex impedance, phasor diagrams, RLC circuits, leading vs. lagging
Summary: These activities are meant to help students gain facility with different representations of complex functions, and with relating them to the behavior of an RLC circuit. They begin by plotting voltage (and current) relative to a given current (or voltage), using a specific value for the complex impedance. They can compare their answers with trigonometric representations, and resolve difficulties in deciding whether one function leads or lags the other in time. Students then determine the total impedance in an RLC circuit in terms of the impedance for each circuit element, and plot various vectors in a phasor diagram to see how they are related.
Comments: Students are assumed to have either completed the previous activities on complex exponentials, or had some kind of introduction to writing complex numbers in various forms, and the multiplication of complex exponentials (frequent errors come from not being familiar enough with the rules of exponents). There has been a great deal of confusion among students concerning whether a voltage is leading or lagging the current, depending on which representation being used. It seems to be fairly intuitive for them when looking at the phasor diagrams (as long as they’re clear on the direction in which the vectors are rotating with time); but the trigonometric representations can be challenging because, in the graph of a function that is leading in time, it peaks at a point that is to the physical left of the peak for the function it leads, and therefore “looks” like it’s actually lagging.
Topics: Maxwell-Ampere law, conservation of charge, E- and B-fields for a charging capacitor.
Summary: After first converting the Maxwell-Ampere equation from differential to integral form, students draw conclusions about dp/dt and Div.J for a circuit with a charging capacitor, and compare them with what’s predicted by the static form of Ampere’s law. They are then asked to compare these incorrect predictions with those for the full Maxwell-Ampere equation, and consider how this is related to the continuity of field lines for a divergenceless field (the vector field Curl.B).
Comments: The other activity on this topic (Maxwell-Ampere Part 2, #6B) can also be done in approximately 25 minutes, so the two parts could potentially be used in the same class period, or just split between two classes. When deriving the Maxwell-Ampere law in integral form, 40% of our students incorrectly substituted Qencl/eps_0 for the open-surface flux integral of E (an incorrect application of Gauss’ law, where the flux integral is over a closed surface). Many students were confused about the sign of the net flux of the current density in a region where a capacitor plate is charging – usually because they were not considering the different directions the area vector points in around the Gaussian surface; many were incorrectly thinking that a net charge flowing into the volume would correspond to positive flux. About 1/4 of our students were confused by the questions regarding charge conservation, thinking they were instead asking about whether there was an equal but opposite amount of charge on the two capacitor plates. In a handful of cases, students initially believed that charge was actually flowing through the space between the capacitor plates, so that the charge flow was continuous through the circuit. Students may need to be reminded that the divergence of the curl of a vector field is always zero.
Topics: Maxwell-Ampere law, Gauss’ law, E- and B-fields for a charging capacitor, B-field of a current-carrying wire.
Summary: After converting the Maxwell-Ampere equation from differential to integral form, students find the magnetic field outside a current-carrying wire. They then consider the electric field between the capacitor plates in terms of the current in the wires and the charge density on the plates, and derive a formula for the magnetic field between the plates induced by the changing electric field. They can then compare the magnitude of the field outside the wire with the field between the plates, specifically for the case where the radius of the Amperian loop is such that it encloses all of the electric flux (they are then the same).
Comments: The task of converting Maxwell-Ampere from differential form to integral form is repeated because students have shown they have real difficulty in doing this without a textbook in front of them. Instructors may not want to skip this if Parts 1 & 2 are used in the same day. Many students showed continuing difficulty with choosing the correct surfaces and loops for applying the integral equations – specifically with seeing how the two types of integrals are related to each other by the same surface. An additional task for students could be to explain the final result in terms of the continuity of field lines for the divergenceless field Curl.B, which was also addressed at the end of the previous tutorial.
Topics: Boundary conditions, Maxwell’s equations in integral form.
Summary: These activities guide students through a derivation of the boundary conditions on the electric and magnetic fields at the interface between vacuum and a general material. Initial tasks have them consider the charge/current/flux enclosed by imaginary surfaces. They are then guided to apply Maxwell’s equations to solve for the conditions on the fields at either side of the boundary.
Comments: Just prior to implementation, we gave our class a brief review of sign conventions regarding unit vectors and integration surfaces/loops. During the tasks, many students were still introducing minus signs into the equations from memory (or intuition), without being able to justify them in terms of the direction of the unit vectors. Some students are confused by the distinction between a surface current and the volume current “right at the very edge” of a material, and this is addressed by having them consider the charge/flux enclosed by surfaces with dimensions that shrink to zero, in this case just across the surface. Some students got very caught up on whether the charge/current/flux enclosed is actually zero, or just vanishingly small – this can be an opportunity to discuss comparisons between finite quantities and ones that are differentially small. We have purposefully avoided reference to the auxiliary fields D & H, and there is no distinguishing here between free and bound charges/currents, because of the added complexity this would involve.
Topics: Poynting vector, boundary conditions, surface charges, Ohm’s law
Summary: These activities focus on the location and direction of energy flow for a circuit containing just a battery and a resistor; the initial tasks consider only a resistive element with a current flowing through it. Students should first conclude that energy is flowing radially into the resistor (and not along the direction of current), and that Faraday’s Law requires the electric field to be nonzero outside the resistor. With no volume charge density inside the resistor, the next conclusion is that surface charges are responsible for the perpendicular components of the electric field, which must vary along the length of the resistor for the field to be conservative. The final conclusion is that energy flows from battery to resistor through the fields outside the conducting wires, and that energy can (counter-intuitively) flow opposite the direction of current.
Comments: The part concerning the parallel components of the electric field outside the resistor may be more challenging for students who did not complete the tutorial on boundary conditions. There were still a few students who believed the volume charge density inside the resistor is non-zero (even though the current is steady), so this activity gives another opportunity to address this (see #2-Ohm’s Law). The final conclusion about the location and direction of energy flow was surprising to most students, and instructors should be sure that students don’t automatically assume the direction of energy flow is the same through the entire circuit (from positive to negative terminal, instead of outwards from both). Many students strongly associate the Poynting vector only with electromagnetic waves, so this activity provides another context for them.
Topics: Linear differential operators/equations
Summary: Linear operators are defined, and students must determine which of five operators are linear. The second part addresses how the components of a complex solution are themselves solutions to a linear differential equation.
Comments: Students should be sure to check their answers to the first part, since many will mistakenly believe that option IV is linear if they don’t think too hard about it. The final task is designed to address potential confusion about how to translate between complex exponential representations and physical solutions.
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Topics: Wave equation, Maxwell’s equations.
Summary: This is a mostly mathematical exercise, to have students derive the wave equation for the electric field in a vacuum (where there are no charges or currents).
Comments: The initial task of deriving the wave equation should be completed within 5-10 minutes, though some may need help in getting started. The final part asking about static fields has been added since the implementation in our class, but we expect that there will be some students who are confused about whether this statement about fields in a vacuum is general. The biggest confusion we’ve seen for students is how the wave equation for EM fields is usually written as a compact vector equation, where it can look as though the Laplacian is operating on the entire vector, instead of each of its components separately.
Topics: Plane waves, complex exponentials
Summary: The initial tasks have students identifying the various quantities that go into a plane wave represented by a complex exponential; the first involves constructing an expression from the quantities given, while the second analyzes the quantities for an expression that’s given to them. The remaining tasks involve deriving explicit expressions for the divergence and curl of a plane wave, and relating them to Maxwell’s equations in vacuum to determine the orientations of the electric and magnetic fields relative to the direction of propagation.
Comments: We’ve noticed that students can have trouble parsing out the various vectors and scalar quantities that go into a plane wave expressed in complex exponential notation; the first two tasks give them practice with this. When taking partial derivatives of the complex exponential, many students had difficulty correctly applying the chain rule; in particular, they often didn’t see that the dot-product k.r is compact notation the sum of the products of the components - sometimes because they were associating the r-vector only with spherical coordinates. The final page asks students to make a convincing argument for how the electric field is related to the magnetic field in terms of a cross-product with the wave vector – several students were initially trying to calculate the actual cross-product using determinants, without recognizing that the divergence and curl operations just replace the spatial derivatives with the corresponding components of k.
Topics: Reflection and transmission, boundary conditions, complex exponentials.
Summary: Students begin by expressing in exponential notation the boundary condition on the parallel components of an EM plane wave for normal incidence at the interface between two media. They can then find the phase shift and amplitude for the reflected wave by considering representations of the electric field in the complex plane, first for when the amplitude of the transmitted wave is smaller than for the incident, and then when the opposite is true. Students should conclude that the frequencies of all three equations must match for the boundary condition to hold at all times. A second boundary equation is found by considering the electric and magnetic fields of the reflected wave. The remaining tasks connect the amplitude and phase shift of the reflected wave with the refractive indices of the two materials.
Comments: Warning (!): Portions of this tutorial have not been validated or field-tested, but we expect students to be able to finish the tasks in less than 50 minutes. A somewhat different version was used in our class, and the tasks related to representations in the complex plane are new. We anticipate that this way of representing the electric field will be more intuitive for seeing how the electric fields must match up in order for the boundary condition to be satisfied at all times. A number of students will have issues with the algebra when solving two equations for two unknowns on the final page. Checking their answers for the case when the refractive indices are equal may help them to see whether they have it right.
Topics: Reflection and transmission, boundary conditions, complex exponentials.
Summary: Students begin by expressing in exponential notation the boundary condition on the parallel components of the electric field for an EM plane wave at oblique incidence to the interface between vacuum and a material. After finding the phase shift for the reflected wave, students should conclude that the components of k for each of the waves that are parallel to the boundary must match if the equation is true all along the boundary. The remaining tasks connect the angles of reflection and transmission to index of refraction for the material.
Comments: Warning (!): Portions of this tutorial have not been validated or field-tested, but we expect students to be able to finish these tasks in around 30 minutes. A somewhat different version was used in our class, and the tasks related to representations in the complex plane are new. The tasks in this tutorial have been constructed with the assumption that students have completed the tutorial on R&T for normal incidence (#12A); if not, the more abbreviated tasks herein will be more challenging, since they are not scaffolded in the same way as in the prior tutorial. The vectors in the diagrams all have the correct proportions, so it is important that students can justify their answers on the final page in terms of the reduced wave speed, and are not simply judging from the diagram.
Topics: Time-dependent potentials and fields, gauge transformations
Summary: Students are first reminded of why EM fields can be written in terms of a scalar and vector potential. They then show that a gauge transformation in the vector potential results in an identical magnetic field. Students derive an integral relationship between E & A, and then find the necessary conditions for transforming V. A challenge question asks students to derive Poisson’s equation, which would be used to solve for the scalar function λ that transforms the potentials to the Coulomb gauge.
Comments: The biggest complaint from students has been about not entirely understanding why we would want to transform the potentials in the first place. This is hinted at in the final challenge question, where an equation is found for the function that transforms to the Coulomb gauge, but is not explicitly addressed here. The first task is a review intended to orient students to the remaining tasks, reminding them of why we can write the fields in terms of potentials. Many students have forgotten that the various statements that can be made about divergenceless (or irrotational) fields are all equivalent. [See Sect. 1.6.2 in Griffiths] Some students were unsure about the cross product being a linear operator (whether the curl of the sum of two vectors is equal to the sum of the curls). There has been a modest amount of confusion in simply keeping track of primed and unprimed quantities, with students sometimes mixing them up in their heads.
Topics: Retarded time, time-retarded potentials and fields.
Summary: Students explore the concept of retarded time for the case of an infinitely long wire with a current that abruptly starts at t = 0. They first consider the points in space where an observer would be aware of there being a non-zero current a short time after it starts. Students find that the retarded time has different values at different points in space (relative to an observer at the origin), and decide on the limits of integration for finding the retarded vector potential at the origin. Challenge questions at the end have students calculate the electric field at the origin, and check their answer in the limit of long times.
Comments: Although it is fairly intuitive to students that it takes a finite amount of time for effects to propagate from a source to an observer, the definition of retarded time (and how it is used in a calculation) is not. In particular, that the retarded time is a function of two coordinate variables, and has different values at different points in space relative to a fixed observer. There is a “time ruler” on a separate handout that students can use for the questions on the first and second page – they may need a gentle reminder that it’s easiest to work with whole numbers for the distance from the origin (some were tempted to estimate the distance for points on the wire that were even with the tick marks on the x and y-axes). There were some students confused about what the primed and unprimed variables are each referring to, which typically shows up in problems involving the separation vector r - r'. The tasks in the challenge questions at the end are similar to Example 10.2 in Griffiths, but a simpler method (involving the fundamental theorem of calculus) is used for calculating the electric field from the retarded vector potential.
These are all relatively short activities, meant to address just some of the basics from special relativity, such as length contraction, 4-momentum, and velocity addition.
Topics: Special relativity, Lorentz transformations, length contraction, simultaneity.
Summary: Students first establish the relationships between the times and locations that go into the length measurement of a moving body. They then derive a formula for length contraction using the Lorentz transformations, and consider whether the two position measurements occur at the same time in both frames.
Comments: Although some of the questions may seem trivial to instructors, we found that a number of students were confused on even the “simple” tasks, which shows that students may use the Lorentz transformations without understanding exactly what the different primed and unprimed variables correspond to. Our students were told beforehand that length measurements involve a simultaneous determination of the positions of the two ends; still, some were very tentative about simply saying that the two times are equal, or even that the length is simply the difference between the two position measurements.
Topics: Special relativity, 4-momentum, relativistic collisions.
Summary: Students use conservation of relativistic 4-momentum to find the final mass of an object resulting from the merging of two colliding particles.
Comments: This activity was straightforward for most students, as long as they were clear on the following: definition of relativistic 4-momentum; the total momentum of a system is the linear sum of the momenta of the particles; and that this quantity is conserved before and after the collision. Some students may momentarily forget the velocity dependence of γ when first working out the total momentum; the spatial velocities of the two particles cancel, but the γ-factor that appears in the total momentum is not also zero.
Topics: Special relativity, Lorentz transformations, relativistic addition of velocities.
Summary: Students derive the velocity addition formula using the Lorentz transformations and the definition for the velocity in two different inertial frames.
Comments: The biggest difficulty for students may be the algebra involved. A common problem is for students to be confused about the velocity of the frame v, and the velocity of the particle u in that frame of reference. We have also noticed some conceptual difficulty for students regarding an event taking place at a single point in spacetime, and the different coordinate representations of that point in different inertial frames.
E&M II Learning Goals
These learning goals for upper-division electrodynamics were created by a group of physics faculty from a number of research areas, including physics education research. Rather than addressing specific content to be covered in a course (as with a syllabus), this list of course-scale learning goals represents what we think students should be able to do at this stage of their development as physicists. The list of topic-specific learning goals reflects the knowledge and skills that were emphasiszed in the recent transformed E&M II courses at CU Boulder, organized according to their order of presentation in Griffiths.
Griffiths Ch 7: (Maxwell’s Equations, EMF, inductance, boundary conditions)
Students should be able to...
Griffiths Ch 8: (Conservation laws for charge, momentum & energy)
Students should be able to...
Griffiths Ch 9 (9.1-9.3): (Electromagnetic waves, reflection and transmission, dispersion, EM waves in conductors)
Students should be able to:
Griffiths Ch 10: (Potentials, gauge invariance, retarded time)
Students should be able to...
Griffiths Ch 11: (Radiation) Students should be able to...
Griffiths Ch 12: (Relativity) Students should be able to...
(Includes publications for both E&M 1 and 2)
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This material is based upon work supported by the University of Colorado, The Hewlett Foundation, and the National Science Foundation under Grant Numbers DUE 1023028, DUE 0737118, PHY 0748742, and CAREER 0448176. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.