### Model vs mechanism -- trouble between the sheets

As many of you who know me are aware, during this academic
year 2011-12, I’m deeply involved in an attempt to reinvent intro physics for
Life Science students. Our traditional approach for this population – which
includes prospective biology researchers and pre-health-care professionals –
tends to look like a traditional class for mechanical engineers cut down in the
math somewhat with a few “spherical cow” problems tossed in. Our group is
trying to see how to make it feel more “biologically authentic” for our bio
majors and pre-meds.*

I’m teaching a small (N=20) trial class and I’m teaching it
in what’s being referred to this year as a “flipped class”. Instead of the
students getting the basic material presented to them in a lecture and then
going out to do problems on their own, they are expected to get the material
themselves (typically, watch a lecture video) before class and then go to an
“active engagement” class with problems and discussions. I like this idea,
since it puts the effort into where on-site education adds value over web
instruction – in live interaction.**

I’m not preparing lecture videos – rather, our team is
preparing materials as a wiki-book. The students read two or three web pages
the night before class and write a paragraph summary of each page and ask a
question. I answer their questions on-line before class and use the best
questions to open discussions in the next morning’s class. I think this has an
advantage over video in that they are learning to read scientific text – and
not just read it but evaluate and question it.

This experience is reminiscent of the famous old EDS superbowl
commercial, “Building an
airplane in flight.” The material for each class is only ready for the
students 36 hours before class (if I’m lucky). Each evening before class I have
to come up with at least half a dozen clicker problems to guide us through the
critical elements. Then during each class I have to be on my toes, looking at
the students’ answers, listening to what they are saying –and actually hearing
it! And I have to be able to respond in the moment. No more, “That’s a very
interesting question. I’ll bring you the answer tomorrow.” Rather, I now try to
respond. “Great question. What do you all think?” This definitely started out being
out of my comfort zone. Although I’ve been giving clicker lectures for 10 years
to classes of 200, I’ve always told them not to bother reading a text, and
spent much of my class time giving them clearly outlined core principles and
then doing a few examples with 3-4 clicker questions. This new class is an
adrenaline high and I think I’m getting addicted to it. In any case, my new
teaching is not the point of this entry – what I learned from last week’s
classes is.

Two of the broad threads that run through all of physics are
the following. First, some advice I give to my students on day 1:

*Physics is about something real. Whenever you think about a physics example, start from a mental image of a physical situation and refer everything back to it.*

In physics our equations, our graphs, our diagrams, are all about
some physical situation and are intended to inform us about some aspect or
relationship of that situation, building a richer and more multi-dimensional
view of it. In my long experience as a physics teacher (40 years), this element
is what’s missing for most intro students. They want to “answer-make” rather
than “sense-make” and fail to build that underlying physical picture. This is
really what we are asking them to do when we advise them, “First, draw a
picture”, but they often do it and then don’t use it for anything. Having a
physical model can (and should) guide them in understanding the mechanism of what’s
happening, in eciding what they have to pay attention to and what they can
ignore, and in figuring out what principles are relevant with what
restrictions.

Second, I tell my students:

*The “style” of physics is to simplify. We always try to find the simplest example that illustrates a principle so we can understand it fully. We then use that example as the core of our thinking to elaborate into more realistic situations.*

This is what I take to be the essence of the famous Einstein
quote – “Physics should be as simple as possible, but not simpler.” It’s our
motivation for doing point masses, flat-earth gravity, the ideal pendulum and
the perfect Hooke’s law spring. (And the spherical cow.) It’s why we spend so
much time, both in our instruction and in our research, in what we sometimes
refer to in a self-derogatory but affectionate way as “toy models.” It’s our
way of getting a foothold that we can make sense of to imbed into organizing
and “finding the physics” in a complex situation.

These two characteristics of physics lead me to two
fundamental goals of my physics instruction: (1) help students put that
physical picture into their every analysis of a physics problem, and (2) help
students understand the simple toy models and learn how to use them to organize
their physics thinking.

Last week I had a dramatic example of how there is a dynamic
tension between these two aspects of physics thinking. We were studying
electric fields and potential. A standard example is the “infinite flat sheet
of uniform charge.”*** This is a nice example since the math simplifies
dramatically. Because of Coulomb’s law, any electric field has to look like a
charge divided by the square of a distance (times a universal constant chosen
to set the measurement scale). When we have an infinite sheet, we have no “charge”
we can use – it’s infinite –we only have the charge per unit area. This already
has units of charge divided by length squared, so there is no room for any
other distance in a formula for the electric field. The result is that the
field has to be a constant, independent of the distance from the sheet.

This seems strange, but it actually makes sense. [This
argument may be difficult without a picture. To see details with figures, go to
our draft
webpage on the field of an infinite sheet, but I’ll outline the argument
here briefly. You can skip to the next paragraph if you don’t want the
technical details. They’re cute but not essential to my point.]

To get the total effect of the infinite sheet you have to
add up the Coulomb’s law contributions from each of the bits of charge in the
sheet to the field at the point you are sitting at. Each bit of charge contributes
a field vector that points along the line to your point from the charge that is
proportional to one over the distance to that charge squared. As you go farther
away to the more remote charges, they contribute less and less. Also, each
distance charge is paired with another distant charge equally far away on the
other side and these contributions tend to cancel – and cancel more and more
the farther away you get. The result is that for the entire infinite sheet, if
you are a distance s from the sheet, only the circle right beneath you of
radius about 5s contributes significantly to the field you detect. So although
we say we have “an infinite sheet” that’s not what we mean. We mean: we have a
flat sheet and the edges are far enough away that we don’t have to worry about
them. So the result is: only a circle of radius 5s matters. As you increase s,
the effect of each charge on the sheet falls off like 1/s

^{2}, but the amount of charge you see grows like s^{2}. These two effects cancel to result in a constant field.
The result of having a constant E field simplifies a lot of
the math. The potential that goes with a constant field is just linear (since
the derivative of the potential is the E field) so the math is really simple- 9

^{th}grade algebra. All those complex “curvy 1/r^{2}” functions and vector integrals add up to give straight lines. It looks just like the same math for flat-earth gravity – where we take the gravitational field to be constant always pointing in the same direction.
So here’s where I ran into something interesting last week.
Students read my draft webpage (and it was read and summarized by 20 out of 20
of the students). Then in class I asked the following clicker questions:

*If two uniform sheets of equal and opposite charge can be treated as if they were infinitely large, which of the following graphs might serve as a graph of (A) the x-component of the electric field and (B) the electrostatic potential as a function of the coordinate x along the dotted line?*

Since the fields from each sheet are constant and since the
charges are equal and opposite, outside of the sheet the two fields cancel, and
between them they add. The result is that the E field (x component) looks like
graph 8; constant between the sheets and zero outside. Looking for graphs whose
(negative) derivative looks like 8, we see that both 3 and 9 work for the
potential. That’s OK since for the electric potential (as for height when we
are talking about gravity) we can choose any reference point as zero potential.
It’s only the shape that matters physically.

On the first question – what does the electric field look
like – the results were gratifying. Of the 19 students present, 16 chose answer
8. But one student complained. He said, “I didn’t like any of them.” When I
asked why, he responded, “Because when you get near to the sheets you’ll see
the individual charges and the field has to go to infinity.” I brushed him off
with a brief comment about resolution – that it would only happen really really
close and we wouldn’t see it on this scale and anyway we were ignoring
individual charges in this simple model.

On the second question – what does the potential look like –
the results were more mixed. About half chose answer 3, but the rest were all
over the lot. But one student (a different one) said, “It can’t be any of
them.” When I asked why, he responded, “Because if you are sitting exactly
between the plates the result has to be zero. If you are sitting there, for
every positive charge on one sheet there is a negative charge on the other
sheet an equal distance away that will cancel. Also, it has to eventually asymptote
to zero for large values.”

Well! I was stunned. This was absolutely top quality physics
reasoning. He was using a physical picture and using it with a correct symmetry
argument – another strong tool in the quiver of good physics thinking. (This
was something I had been trying to model in the class, but not fussing too much
about.) Finally, he was focusing on limiting cases, another standard tool we
try to get our students to use. My first internal response was – sign this guy
up as a physics major! My second was. Gak! I seem to want them to be looking at
this model example NOT in terms of the basic physical elements but as a toy
model that suppressed the underlying physical picture. Since a third major goal
of my class is to teach my students to

*seek consistency*, what was I doing?
I don’t know that I handled my little epiphany well on the
spot. I tried to be encouraging and praise the students’ thinking but still bring
us back to being able to using the simplified model while “suspending our
disbelief” – using it even though we know that “when we go to infinity the
sheets that look infinite when we are close will turn into looking like point
charges when we are far away”, and “we’ll ignore the fact that the charges are
actually quantized and treat the chunky peanut butter as if it were perfectly
smooth since we don’t see the individual charges until we are really close and
then we’ll just ‘close our eyes’ for a moment while we are passing through the
sheet”.

Really what it’s about is modeling.**** When we have complex
situations – like lots and lots of charges – it’s great to have “a stake in the
ground”; an example where we know an “exact” answer that we can refer to that
serves as a starting point for further examples and elaborations. But in intro
physics we tend to focus on the simplicity of the models and not on the
complexity of thinking about where they come from, what their validity is, and
what value we can make of them.

Increasingly, I want to open up this Pandora’s box for my
students. Trying to pretend that the physics is simple by hiding the deep
structure, both ontological and epistemological (i.e., what is it we are
actually talking about and how is it we decide we know), is beginning to seem
to me to be unfair to our students and not the best way to start students on
learning physics.

* For more info on this, check out the NEXUS physics
class website.

** If you haven’t already listened to Emily Hanford’s audio
documentary, “Don’t Lecture Me”, check it out at American
Radioworks.

*** I’m referring to the example as “sheets” rather than
“plates” as is more common since my examples for biologists will be membranes
rather than parallel plate capacitors with metal disks.

****David Hestenes, "Modeling games in the Newtonian
World",

*Am. J. Phys.***60**(1992) 732-748.