Tutorials & In-Class Activities

The 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 generally focused on either promoting an understanding of important topics from second-semester E&M, or guiding students through derivations that would typically be done during lecture. They were often inspired by in-class observations of student difficulties, and have been tested in focus-group student interviews (designed to mimic a tutorial setting) and in the classroom. We provide here information about how they were implemented, and a summary list of the activities and the topics they cover, including an estimate of the amount of class time they require. **Clicking the title of an individual tutorial in the sidebar at left takes you directly to its summary below.** You can also download a PDF version of this tutorial guide.

**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).**

If you are interested in downloading **just** the short in-class activites (requiring from ~5-30 min. each, briefly described here) from the FA11 semester as a single package, this is available at left, though these are also individually available through the course calendar, where you can see the date and context in which each was assigned.

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

I. Notes on Implementation |

II. Summary List of Tutorials |

### I. Notes on Implementation

All of 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 the “usual” tutorial setting, 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 the CU SP12 lectures were partly or fully replaced with tutorial
activities. We typically oriented students to the activities before they began with a concept test and/or
discussion. Otherwise, they were implemented at appropriate times during lecture, most often during the middle, sometimes
at the very 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.[1]
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.**

### II. Summary List of Tutorials

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 have tried to make note of this in the summaries 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, followed by a brief summary of the tasks involved, and abbreviated
comments on their implementation (including some common student difficulties). The pdf files in this package contain
the complete notes at the beginning of each activity. **Clicking the title of an individual activity in the list below takes you directly to its summary.**

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. |

### 00 – Review Material

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.

### 0A – Divergence and Stokes’ Theorems (~15 minutes)

**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.

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### 0B – Gauss’ Law (~15 minutes)

**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.

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### 0C – Ampere’s Law (~15 minutes)

**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.

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### 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.

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### 02 – Ohm’s Law (< 50 minutes)

**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** = σ**E**). 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.

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### 03 – Faraday’s Law (< 30 minutes)

**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.

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### 04 – Complex Exponentials (< 50 minutes)

**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).

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### 05 – Complex Impedance (< 50 minutes)

**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.

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### 6A – Maxwell-Ampere Law, Part 1 (~25 minutes)

**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.

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### 6B – Maxwell-Ampere Law, Part 2 (~25 minutes)

**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.

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### 07 – Boundary Conditions (< 50 minutes)

**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.

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### 08 – Energy Flow in a Simple Circuit (< 40 minutes)

**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.

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### 09 – Linear Operators (< 15 minutes)

**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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### 10 – EM Wave Equation (~10 minutes)

**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.

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### 11 – Complex Plane Waves (< 40 minutes)

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**.

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### 12A – Reflection & Transmission (Normal Incidence) (< 50 minutes)

**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.

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### 12B – Reflection & Transmission (Oblique Incidence) (~30 minutes)

**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.

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### 13 – Gauge Invariance (< 50 minutes)

**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.

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### 14 – Time-Retarded Potentials (~30 minutes)

**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.

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### 15 – Special Relativity

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.

### 15A – Length Contraction (< 15 minutes)

**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.

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### 15B – Inelastic Collision (~10 minutes)

**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.

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### 15C – Relativistic Velocity Transformation(~10 minutes)

**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.

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