E&M Modeling: Flux part 2

Setting the Stage

In my last post I wrote about a theoretical approach to having students discover the concept of electric flux for a point charge. While they struggled with explaining what the proportionality constant for their Electric field vs. enclosed area graphs meant (aka, the flux), I was still satisfied with the results. What turned out to be the most useful question was asking them why I might have chosen a spherical surface to draw around the point charge as opposed to some other shape. After all, it’s a completely imaginary surface, so why this one?

They didn’t have much to say until we spent some additional time refining the equation we’d come up with:

fluxequation2

Those of you more familiar with the concept of flux will recognize the missing cos(theta), so my next goal was to have them choose to add that term to the equation.

I started the discussion by holding up a whiteboard and told them to imagine a steady stream of wind flowing with a constant speed directly at me. I asked At what angle should I hold the whiteboard so that I could “harness” the wind stream as best I can?

It only took a few moments for them to realize that angling it completely perpendicular would allow them to maximize their “use” of the wind stream and that holding the board parallel would minimize it. I then asked about the wind then blowing at some in-between angle and put this picture up to help them visualize.

flux2

They first decided that this would yield some level of “usefulness” that was in-between the maximum and minimum values based on the angle between the vectors and the surface. Asking which “part” of the vector contributed to the “usefulness” led them to conclude that the perpendicular component of the vector was the only part that contributed. They were unable to see a difference between using the sin of the angle between the vertical and the vector and the cos of the angle perpendicular to the surface, which is fine because there isn’t one. I said that we should use the angle with the perpendicular only because that’s the convention, which will come in handy later. Their equation was now this:

flux3

Revisiting my original question

My original question was this: Why did I choose to draw a spherical surface around the point charge? Why not any other surface?

I put this picture back up on the board to help with visualization. I’m discovering more and more that the difficulties with flux and Gauss’ Law are primarily with visualizing the various surfaces and charge distributions, so any chance I have to aid in that I take.

fieldlines

They quickly determined that the field lines were perpendicular at each point of the circular surface, which was easily extended to a spherical one. Since each point was perpendicular, cos(theta) is just 1, thus the flux calculation is almost trivial.

Making the flux calculation trivial is one of the primary ways to effectively utilize Gauss’ Law. My purpose here was to continue their development of seeing spatial symmetries and using those symmetries to simplify otherwise complicated calculations.

Finalizing the flux equation

The equation they’d come up with was more generalized, but still not as general as it needs to be for both using and understanding Gauss’ Law at a deeper conceptual level. I also continue to emphasize that the more general a scientific model is, the more useful it becomes. So the issue with their equation is that it assumes a surface in which the field lines are perpendicular at each point of the surface. What if that’s not possible? How can we generalize the equation to work for any surface?

I drew a square around the point charge to help jump start the discussion.

flux4

The question now is this: How can we calculate the flux through the square?

I’m careful to clarify that I’m not looking for them to jump to a mathematical equation, but rather I want them to brainstorm on a conceptual way to do it. We’ll worry about the mathematics later. Just talk me through how you might calculate this.

I should mention that this isn’t the first time I’ve had them do this kind of exercise. I first exposed them to this when calculating electric fields of extended bodies, pre-flux/Gauss’ Law as a way to get them thinking in “calculus-mode.” Even though all of my students are in AP Calc BC, they haven’t gotten to integrals yet, but that doesn’t mean they can’t start “thinking calculus.” Writing this post now has me thinking that I’ll write up my approach to that. So I’ll do that at some point.

Anyway, when they got stuck, I asked them how they could get the contribution to the flux due to an individual field line and then reminded them that the total flux was the contribution due to all the field lines. They came up with this:

We just need to know the E at each point on the square, multiply it by the cos of the angle perpendicular to the square, add all that up, and then multiply by the area of the square.

I then asked them to write that out mathematically yielding this:

flux5

While that may cause a mathematician to cringe (and the Asquare part isn’t fully correct), I wasn’t worried about the correctness of their notation. The essence is there. This also let them see if E was the same at each point of the square and if cos(theta) was always 1, then those things could be pulled out of the sum, which will be useful once they get to Gauss’ Law. I then introduce the more formal definition of flux and made sure to show how each element of their equation translated to this definition:

flux6

I also briefly discuss the dot product, though I honestly haven’t done a great job of helping them build their understanding of it. I’m not sure that I care right now… but we’ll see if that comes back to bite me later one. I’m careful to emphasize that the only real differences between their equation and this one is that of notation. I explain what the circle on the integral means along with the S below the integral.

Next up: Gauss’ Law

My students still weren’t exactly sure why we we were spending so much time on this concept of flux. I’d mentioned several times that this would help make deriving expressions for electric fields much easier than the brute force method they’d done previously, but that doesn’t mean they see why this will be a path to that.

One student had alluded to looking at the relationship between the flux and the charge inside the surface, which is exactly what Gauss’ Law is, so I brought that point back up for discussion. In my next post, I’ll talk about how I had them approach investigating that and how they used it to come up with Gauss’ Law.

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E&M Modeling: Flux part 1

The Dilemma

It’s no secret that I’m all about Modeling Instruction. While I don’t necessarily follow their curriculum to the “T”, the ideas behind it undoubtedly form the backbone of almost everything I do in my class. And I definitely use a fair share of their worksheets and labs almost verbatim.

The basic idea behind modeling is that all the equations, rules, laws, etc. that are typically delivered to students and expected to believe a priori are instead derived and discovered through a series of paradigm labs performed at the beginning of an instructional unit. This works great when you’re rolling balls down ramps or swinging pendulums, but it’s much more difficult (if not impossible) to this with things like Gauss’ Law or Electric Flux given even my very well funded equipment budget.

Thus, my dilemma. I’m teaching AP Physics C: Electricity and Magnetism for the first time this year, and I was terrified that I’d be reduced to mostly lecturing. I hate lecturing. My kids hate it, especially given that I had them last year. They’ve got expectations of what “Physics with Mr. Register” is like. And it just doesn’t work. Thankfully, desperation is a great motivator for me, and I think I’ve stumbled on to some excellent alternatives.

I won’t pretend to have all the answers right now, but things are turning out quite differently than I planned. The next two three posts will be two different paradigm “lab” ideas that I’m using to have students invent the concept of electric flux and to derive the simplest case of Gauss’ Law.

Where They’re At

My students have just finished working through deriving the electric field due to a thin ring, thin disk, and a half-circular ring the brute-force way:

efield

Which is terrible, even for the most symmetric of situations (well, I think it’s super cool…). I’ve got my reasons for them doing this… but that’s for another post.

So at this point they’re aching for something simpler.

Electric Flux

We start with a quick review of field lines, in particular that the density of field lines at any particular point corresponds to the field strength. I then draw field lines for a point charge on the whiteboard and draw a few concentric circles around the charge.

fieldlines

When prompted with How does the density of field lines change as the radius of these imaginary circles increases?, they quickly reply that the density decreases, which means that the field strength also decreases. Which is nothing new because we’ve been dealing with electric fields for 2 weeks now.

I then tell them that we’re going to examine the relationship between the area of these imaginary circles and the density of field lines within that circle, i.e. the field strength. Unfortunately, my only response to the question Uhh… why? is that they should trust me because it will take the nasty integral away. And because why not?

Except that we’re going to do it in 3D with the surface area of a sphere to better model reality, us living in a 3D world.

flux

Instead of setting this up as an actual experiment, I explain that we’re going to put our theoretical physicist hats on.

They’re tasked with calculating the electric field strength of a 1 nC point charge at a distance of 1 to 7 m (in 1 m increments) along with the surface area of the accompanying sphere.

fluxData Once they’re done with their calculations, they dutifully jump to graphing the data:

fluxplot

Linearizing and slope finding occurs, and then they generate the following equation:

fluxequation

After some discussion as to the meaning of the slope, students come up with essentially the definition for electric flux: the “amount” of electric field penetrating a given area. Then some rearranging and symbol assigning, and we’ve come up with a (very limited) mathematical definition of electric flux.

fluxequation2

I also pose some additional, fairly standard follow-up questions:

  • How does the flux change as the field strength increases and decreases? The area?
  • How does the field strength change as the flux increases and decreases? The area?

Nothing very difficult at all for students in AP Calculus BC, but still important to discuss.

What’s Next?

We need to now generalize the equation for electric flux to include the pesky “only the perpendicular parts of the field are important” condition. I’m still figuring out how to have them arrive to that conclusion… but my current idea is to pose the following question: Why did I choose a sphere? Why not some other shape? Hint: think about symmetry. 

Ideally, they’ll see the spherical symmetry of the situation and that will shed some light on why the fact that the field lines are perpendicular to the surface of the sphere at every point is important.

After a day or two whiteboarding with this new idea, I’ll ask students to investigate the relationship between the flux and the charge enclosed within the spherical surface, which will yield the simplest case of Gauss’ Law.

But I’ll wait to post about that once I get there. Until then, if you’ve got any ideas on how to improve on this, please share!

See part 2 here!

Sticky Tape Electrostatics

The Rundown

I’ve been taking a more inquiry-based, qualitative approach to labs and demonstrations. Based on my readings of Knight and Arons, assuming that students have any conceptual understandings of the core concepts in electrostatics would be a mistake. Common alternative conceptions include not identifying charge as a quality of an object, but rather as indistinguishable from the object, not distinguishing between charge and motion of charge, thinking of positive charge as an excess of protons, and many more. Further complicating this is that electrical phenomena is difficult to visualize. What is electrical potential anyway? Like, what is it really? Honestly, I didn’t fully grasp the concepts myself into well after two junior-level college courses in electrostatics and dynamics.

All of this screams at me that it is essential for students to have direct experience with the core concepts of electricity. How can they hope to understand charging by induction or the difference between conductors and insulators (beyond reciting textbook definitions) if they don’t have a solid understanding of what electrical charge is? Electrostatics is so much more than Coulomb’s Law, and it’s my goal to have my students exit this unit with understandings that I didn’t have even after finishing my B.S.

The Setup

I got the inspiration for this from a set of Modeling documents on electrostatics.  I had groups of 2-3 students set up two sets of these scotch tape “sandwiches.” (Click the picture to enlarge).

em tape

The top, base, and bottom are separate pieces of tape with handle-like “flaps” to make it easier to peel them off.

To record data and observations, students used 2′ x 2′ whiteboards that I keep for various purposes like this. I had them split their whiteboards into 4 sections, like this (click the picture to enlarge):

tape lab setup

The purpose of each section is as follows:

  1. Experiment – Which two tapes are you looking at? TT for top-top, MM for middle-middle, BB for bottom-bottom. NOTE: The first picture above uses top-bottom-base. I did top-middle-bottom. Nothing more than a different preference for naming.
  2. Observe – Write observations about each pair of tapes before pulling and comparing.
  3. Predict – What do you think will happen?
  4. Describe – What did you see?

So, now what?

Since this was my students’ first “official” experience with electrostatics, I didn’t have them predict the first experiment. Predictions should be based on something; otherwise, they’re guesses, and I don’t think that carries much educational value.

The first part was simple. I just had the students pull the two top tapes and slowly bring them closer together (sticky sides facing each other). Exclamations such as “Woah!”, “What’s goin’ on here, Mr. Register?!”, and “MAGIC!” instantly convinced me that this was an excellent idea.

em tape2

And for my next trick, Action…at…a…DISTANCE!

Once they were done playing with their top tapes, they wrote in the “Describe” box what they saw. Nothing fancy, just a few words.

Next, they wrote observations about the two tapes. What about the tapes were the same? What about them were different? I’ve found that it’s important to reinforce to students that there are no right or wrong “answers” here. Just write what you observe, and we’ll sort it out later. This can be difficult for students that have been trained and drilled to be answer-driven. However, they all catch on rather quickly! I then went group-by-group and had them share one observation each, and I wrote each of them on my SmartBoard.

To finish up, I had them follow the “proper” order for the middle-middle tapes and the bottom-bottom tapes. That is, take observations, predict, and then describe what they saw.

The Most Important Thing (TM)

What I’ve discovered is that the most important part of labs and demonstrations like this is that students use their observations to make meaningful predictions. A prediction that’s not based on observations and past results is nothing more than a guess. The idea is for me to choose experiments and experiences and to ask the right questions such that students begin building a mental model for explaining electrostatic phenomena. For this activity, the hope is that students would notice that subjecting the tapes to similar conditions produces similar results. The top tapes are both on top of another piece of tape as are the middle tapes. Both sets of tapes repel. Viola!

Part of this process for the students is to also filter out observations that don’t affect the outcomes. I wasn’t careful to make the tapes the same length, and students will notice that. Some students wondered if the direction of the tape handles had anything to do with the results. In true inquiry fashion, instead of giving the answer, I asked students what they thought and how they thought they could test it. This is great for those groups that work quickly and have nothing else to do.

What’s also important is that students discuss and justify their prediction. It’s not enough to just predict, but they also need to explain why they chose that prediction. Having them reason through their predictions is crucial to the model building process regardless of the accuracy of their predictions or reasoning.

Students will want to use words like “static” and “charge.” I avoided confirming or denying their ideas and instead asked them “what’s charge?” or “what about your observations lead to you talk about charge?” Typically, responses would be “well, like charges repel and opposites attract!”, which describes something well enough, but is likely no more than memorized procedural knowledge. Asking “so, then, what is charge?” a few times usually illustrates the idea I want to get across: charge is more than a few memorized facts.

What’s Next?

The experiments they’ve conducted thus far is not enough for students to have constructed a substantial model for electrostatic charge. To follow up, I plan on using these Rutgers Physics Education Group video experiments. The tape activity only showed them like charges and only repelling, nothing else. Part of the goal of the video experiments is to expand upon that.

I will follow the same basic outline: observe, predict. describe. Ideally, students will abstract “peeling tape off of tape” and “rubbing both rods in the same way” to “doing the same thing to two objects” which yields like charges on those objects, which yields similar results. Hopefully, they will extend that and either accurately predict the third video experiment (one rod is positively charged, the other negatively) OR the result will easily “click” with them.This will connect their experience in the lab with previous procedural knowledge of “like charges repel, different charges attract.”

Students will also notice that the tapes, regardless of which one, want to cling to their hands, the table, or anything that isn’t the other piece of tape. They will proclaim that the attraction is due to both objects being oppositely charged, which shows that they don’t yet understand that a charged object can attract a neutral object. Handling that alternative conception, however, is for another day.

After the video experiments, I will be asking students to write down rules for figuring out if two charged objects will attract or repel after watching how they were charged.

How Can I Make it Better?

The most effective improvement I can make is to integrate an extension to this activity that has students performing some procedure that yields tapes with opposite charges. The differences between what they did to tapes that were like-charged and -unlike-charged would need to be clear and easily observable.

I also need to find a way to have students have a more permanent record of their observations. Something as simple as a handout would work. I realized the importance of this halfway through today, and I’m handling it by writing up something myself based on the most common student responses.

There are likely plenty of other ways to make this better. I just haven’t thought of ’em yet! If you’ve got any ideas, please feel free to share!