E&M Modeling: Inductance

This post is part of a series of posts documenting my efforts to keep with a modeling philosophy in my AP Physics C: E&M class where sticking to the Modeling method can often times be difficult. Getting exact measurements on things like electric field strength, electric and magnetic flux, and current changing over time can be difficult or outright impossible without a bunch of expensive equipment. Here’s another way that I’ve figured out how to adapt and keep as best I can do the method through the amazingly useful and 100% free PhET simulations paired with the Hudl Technique app.

Sequencing and setup

This “lab” was conducted after a 2 week introduction to electromagnetism through Faraday’s Law and Lenz’s, which they constructed a conceptual model of by using the Faraday’s Electromagnetic Lab simulation. The remaining time was spent working the typical run-of-the-mill induced current/EMF problems, including my favorite E&M demonstration of dropping a magnet through a copper tube.

I began by drawing a simple circuit with nothing more than a DC voltage source and a loop of wire, and they discussed how the magnetic field within the loops would change as the current and number of loops increased producing a simple drawing like this:

circuit1

I essentially wanted to get them thinking more about the time-behavior of circuits, which we haven’t done a lot of. Most of their experience has been that either the circuit is on or it’s off, no mention of anything in between.

Students then setup a simple RL circuit using the PhET Circuit Construction Kit (AC+DC) Virtual Lab simulation and answered a series of questions mostly aimed at making sure they noticed all of the things I wanted them to notice.

circuit2.png

question1

question2

To accompany question 5, I asked them to model time time-behavior the potential difference across the bulb and the inductor using a diagram I came up with last year. The idea here was to both solidify their understanding of the time-behavior of the potential difference as well as to set them up to be able to explain the results using Lenz’s Law.

circuit3.png

Recreation of what my students came up with.

They were already familiar with the negative sign due to Lenz’s Law within Faraday’s Law, but they viewed it from the context of determining current direction by looking at the direction of the change in magnetic flux. From here, I added to their drawing to help them understand exactly why the bulb started with no difference in potential across it.

circuit4.png

The opposite emf induced by the inductor cancels out what would otherwise be a 10V potential difference across the bulb.

Armed with the beginnings of a conceptual understanding of what was going on in the circuit, they were now tasked with developing a quantitative model for the current and potential differences within the circuit.

Obtaining and analyzing time-dependent behavior

I have neither the equipment, funding, or even the knowledge of what kind of equipment I would need to procure in order to get the kind of data I want. Instead, I had my students use the free Hudl Technique app, which not only turns any smartphone into a slow motion video camera, but also allows you to advance frame-by-frame through the video with the accompanying time-stamp of each frame. They placed an anmeter and voltmeter in frame of their camera, took a video, and then scrolled through afterwards in order to record the current and potential difference across the inductor at various points in time.

circuit5.png

Armed with a notebook full of data, my students jumped into Plot.ly to determine equations for V(t) across the inductor and I(t) within the circuit. Below are their results.

graph1.png

R-squared of 0.9998. Current function I(t) = 0.998 – 0.992*exp(-0.0544*t)

Students quickly realized that the 0.998 and 0.992 were awfully close to the maximum current of 1 A so were easily able to generalize that part of the equation. The 0.0544, on the other hand, was not so obvious. They’ve yet to begin tackling that yet, though. They’re potential difference graphs produced equally solid results:

graph2.png

R-squared of 0.9999. Potential difference function V(t) = 0.0626 + 9.89*exp(-0.541*t)

Like with the maximum current, students saw the parallel to maximum potential difference. They also noticed that the term within the exponential was almost identical, though they’ve yet to delve into what that means.

Finally, I asked them to plot the potential difference vs. the derivative of the current with respect to time as I knew that the slope of this graph would be the definition of inductance. They, of course, don’t know that yet. Taking the derivative of their current function and plotting it with respect to potential difference values gives the following:

graph3

R-squared of 0.9999. Potential difference function V = 182*dI/dt + 0.0762

Hey! It’s linear! And here’s why my students are for now, which we’ll pick up on the next time we meet. Everything after this is my attempt to explain everything out in the way that I’d want my students to see it. It will essentially be a record of me trying to figure out what all this means because I’m still not yet sure what inductance is beyond being able to qualitatively describe what higher or lower inductance does to a circuit’s behavior. But that’s ok because…

But what is inductance really?

Dimensional analysis can be used to figure out that the units of the slope are volt-seconds per amp, which is a little difficult to explain in the typical For every ___ [unit 1], ____ changes by [unit 2] framework that my students (and me) are used to. It can be shown that the units can also be written in terms of tesla-meters-squared per amp, giving units of magnetic flux per amp, which is much easier to understand. Hey! Inductance basically is a measure of the magnetic field per amp that can be generated in an inductor. Neat.

However, that doesn’t quite help determine how the circuit behaves as the slope, i.e. the inductance, changes.

Solving the equation for dI/dt yields the following:

equation

So, as the inductance increases, the rate of change of current decreases, meaning that it takes a longer amount of time for the circuit to reach the maximum current value. Since the current takes longer to ramp up, this means that the magnetic field generated within the inductor also takes longer to ramp up…

After running off and checking with the simulation, it turns out that I’m correct. Hooray! Though, to be honest, since both I and V are functions of time, I can’t quite shake why my logic feels a bit shakey even though the prediction is accurate. Something to think on.

Actually feeling like I understand inductance is a rather new feeling to me, but please let me know if I’m totally missing something here. You’ll do me and my students a huge service!

 

 

A Diagram for Kirchoff’s Voltage Law

Motivation

Students struggle with both a conceptual understanding and the mathematical application of Kirchoff’s Voltage Law (KVL). The highly abstract nature of potential and electricity creates difficulties in students differentiating between and simply understanding such topics as charge, current, potential difference, and power (1), (2). I have developed a diagram that builds conceptual understanding of potential and potential difference and also aids in generating a system of equations adhering to KVL.

While I’ve seen similar qualitative potential vs. position graphs in several common introductory physics textbooks, none of them incorporated multiple loops within the circuit nor were they explicitly discussed as a tool to use in problem solving. My goal here is to address both of those points and have these diagrams to be as integral to circuit analysis as free-body diagrams are to dynamics.

I’ll be presenting this during a poster session at the 2015 Physics Education Research Conference in College Park, MD. Come and find me during then if you’ll be there!

The KVL Diagram

At it’s core, the diagram is a qualitative potential vs. position graph that incorporates all loops within the circuit. Electric potential at any point is represented by the height of the line on the diagram. Different loops within the circuit are represented by different branches on the diagram. To generate a diagram, the following guidelines should be followed.

  • Draw a horizontal dashed line to represent the ground potential, 0 V.
  • Pick a point on the circuit to begin. The battery or other source is best.
  • Upward slanted lines represent the rise in potential provided by a source.
  • Flat lines represent the wire between elements in the circuit.
  • Downward slanted lines represent the drop in potential that charges experience when passing through that element.
  • Choose a single loop to follow and complete each loop one at a time.
  • After the final element in a circuit is reached in a loop, make sure the line drops to the 0 V dashed line.
  • For additional loops, simply extend a horizontal flat line from the first loop at the point at which the paths diverge.

I’ve taken three simple circuits and diagrammed them below to give an idea of how they look. The points (A, B, C, etc) aren’t part of the diagram, but are presented here to help you see how the schematic “translates” to the KVL diagram.

kvl2

Limitations

The diagram’s primary advantage is in generating equations according to KVL without knowing anything quantitative about the circuit. The diagram is meant to be a stepping stone towards such an analysis. Given this, without any quantitative information, the relationship between the magnitudes of different potential differences cannot be known. However, since the height of a line clearly represents absolute potential, students could mistakenly conclude that one potential drop is more, less, or equal to another. In the first example, the drop in height of the line is shown to be the same, but without any additional information, it cannot be known that ΔV1 is equal to ΔV2. Therefore, this limitation must be explicitly discussed with students. Additionally, the slope of slanted lines do not yet provide any information about the circuit.

Student Work Sample

The following picture shows an example of student work using the KVL diagram. This was taken from an AP Physics 1 class during the Spring 2015 semester.

Students worked on the problem below in groups of 2-3. The problem was taken from TIPERs: Sensemaking Tasks for Introductory Physics by  C. J. Hieggelke, Steve Kanim, D. P. Maloney, and  T. L. O’Kuma. Students were asked to rank the magnitude of the potential difference between points M and N. All of the bulbs and batteries in each circuit are identical.

kvl3

Students were not given any of the typical “rules of thumb” for circuits, such as the potential drops across parallel elements are equal. Through working problems like these, this was done to allow students to discover such relationships on their own as well as to be a soft test of the diagram’s effectiveness in developing conceptual understanding. Anecdotally, students found the diagrams useful, especially in differentiating between current and potential difference

Invitation to collaborate

This is only the beginning for me with this diagram. Over the next several years, I will be doing some kind of research on the effectiveness of these diagrams. I’m not yet sure what my research questions will be or what such studies will look like, but I know I’ll be doing something! If you’ve got any feedback on the diagrams or want to participate in research with me, then let me know! Find me on twitter (@TRegPhysics) or shoot me an email!

References

(1) Cohen, R., Eylon, B., and Ganiel, U. (1983), “Potential Difference and Current in Simple Electric Circuits: A Study of Students’ Concepts,” Am. J. Phys. 51, 407.

(2) McDermott, L. C., and Shaffer, P. (1992), (a) “Research as a Guide for Curriculum Development: An Example from Introductory Electricity, Part I: Investigation of Student Understanding,” Am. J. Phys. 60(11), 994.

Electrical Resistance With Drinking Straws

The CASTLE Curriculum

Let me first say that this is a genius idea. Like most of the genius ideas I’ve implemented in my classroom, they’ve come from someone else! Someones, specifically: the Pasco Scientific CASTLE curriculum for circuits (section 2, activities 2.7 and 2.10). I cannot rave enough at how wonderful the curriculum is, even just straight out-of-the-box.

What I love about it is the laser-like focus on conceptual underpinnings and on confronting student preconceptions head on. Discussions in my classroom focus on observations, explanations, analogies, and hypotheticals all supported with a variety of clearly observable physical evidence. I’ll be writing more about the curriculum as a whole once I complete my circuits unit. For now, I wanted to focus on this particular lesson.

Why this idea is genius

One word: analogies.

It’s critical for students to be able to visualize what’s going on in a circuit. Otherwise their knowledge of circuits is restricted simply to how to algebraically manipulate Ohm’s Law and Kirchoff’s circuit laws. Perhaps they know how to calculate equivalent resistance in series and parallel circuits, but that doesn’t mean they have any deep understanding about electrical resistance or current. Unlike much of the mechanics content, however, we can’t literally see what’s happening in a circuit. Instead, we’ve got to use analogies.

This isn’t to say that simply being able to literally see something in physics = deep understanding. Plenty of students go through mechanics seeing all kinds of demonstrations yet still leave their class with only a “formula hunting” mindset (including myself). There’s already plenty of thorough analogies between water flow and circuits out there. Why not just show a few slides demonstrating the water analogy and be done with it? Well…uhh… because it doesn’t work. Telling students what physics is like just. doesn’t. work.

So, what I really should have started this section out with is this.

Three words: student-constructed analogies.

This activity guides students through making directly analogous connections between air flowing through a straw (a “resistor”) and charges flowing through a circuit. It also helps them think more conceptually about resistors in series and in parallel. The best part? I don’t tell any of these connections to my students. They make all of these connections on their own. And that’s why it works!

Setup and background

I did this after about a week of instruction with circuits. At this point, students have established the following:

  1. Something is moving in the wires when the bulbs are lit or when a resistor is connected in a circuit.
  2. What flows in a circuit flows in a continuous loop.
  3. Resistors don’t “use up” electrical current.

Each student is given 4 coffee stirring straws and one regular drinking straw. I made sure to cut the drinking straws to be the about same length of the coffee straws to eliminate that as a variable.

Here are the questions that I have students answer, taken straight from the CASTLE curriculum packet. I skip some of them for various reasons, so here are the ones I do use along with my comments on their importance.

  1. Compare the amount of time it takes you to completely exhale through a drinking straw and through a coffee straw.I let my students use their cell phones to time this. There’s quite a noticeable difference in the times, which is great. I stress to students to do a natural exhale each time and to take about the same amount of breath each time. The “same amount of breath” comes in handy for question three.
  2. What other differences do you see or feel between the two straws? I added this one myself.I want students to use as many observations as they can to justify any explanation. The amount of time it takes to exhale is one solid piece of evidence, but I want more. It also gets students thinking about each individual piece of the experiment, which is useful for the reflection questions later on.
  3. Do you exhale more air through either straw?This question is critical. A common preconception that students have about circuits is that current is “used up” (or consumed, neutralized, absorbed, etc.) to light bulbs or heat up resistors. Since students are taking the same amount of breath each time, the same amount of air is exhaled through each straw. The difference, then, is at what rate the air flows.  this preconception was confronted in a prior activity, and even more evidence (albeit indirect) to dash that preconception is important. Once students conclude that it’s not the amount of air that’s different, I follow up with “well, what is different, then?”
  4. Repeat the activity with each straw, and this time direct the flow of air from the straws into the palm of your hand. What does your hand feel?Another piece of evidence that students can use later to justify their explanations and analogies.

Once students complete this observation phase, I have them answer these additional questions that I came up with.

  1. Which of the straws allows for a greater flow of air? What is your evidence?
  2. Which of the straws has a greater resistance to the flow of air? What is your evidence?

These questions are useful in that it establishes the crucial “facts” and sets students up to draw direct links between this activity and electrical resistance and current. My students had no issues drawing accurate conclusions, and we didn’t have to spend much time discussing these as a class.

Extension questions

Ok, so, maybe I exaggerated a little.

Mind = blown. Typical student reaction to the conclusion of the activity.

Ok, so, that may be a tad bit exaggerated. But only by a bit! Here’s how I get the discussion started:

How is this activity analogous to electrical resistance and electrical current?

Students were puzzled at first. Some weren’t sure what “analogous” meant. Some weren’t sure how to get started. Here’s the prompt I used to help them get going: the ___ in this activity is like ___ in circuits because…

I gave students 8-10 minutes to write as much as they could drawing links between this activity and electrical resistance and current and to share their thoughts with their table mates. I asked each table to share one thing they’d come up with, and I’d write it on my whiteboard. I also encouraged students to write down ideas they saw that they didn’t come up with. The results were excellent. Here are some of the many ideas that I got:

  • The air is like the charges in a circuit. It’s what’s flowing.
  • Your lungs are like the batteries because they’re what’s pushing the charges. This will be used later to tackle the preconception that batteries are the sole source of charges in a circuit. It also sets up an analogy they can draw later between the pressure your lungs exert and potential difference.
  • Air isn’t “used up” by pushing it through the straws, just like charges aren’t “used up” by pushing them through a resistor.
  • Straws are like resistors because the air is getting crammed into the straws and is slowed down. Charges are “crammed” (or “bottlenecked”) into resistors, which slows the flow of current.
  • The “effort” you exert to exhale is like the energy it takes to push charges. This one is fantastic. Most of my classes came up with this one. I’d not even thought of it,and it helped me understand the relationship between energy and current better. Even better is that we’d not specifically tackled this difference in class, and they did it by themselves.

The best part? All I did here was write down student ideas. I didn’t tell them anything. They came up with all the ideas all on their own! I would ask them to elaborate or clarify from time to time, but I kept my own ideas out of it. What’s also cool is that most of the ideas they came up with I hadn’t thought of, and each class contributed a new idea that a previous class hadn’t thought of. Students loved hearing from me that they were teaching me something! I was absolutely blown away by this! It’s reinforced my general belief that our society doesn’t give kids enough credit. They’re deeply creative and capable of brilliance at any age.

Reflecting on the activity

Though I can’t quote it exactly, I’ve seen Rick Wormeli write that learning doesn’t happen during an activity; it happens when students reflect on the activity. This activity brought that wisdom to life for me. Without the class discussion or questions prompting students to make connections for themselves, the activity isn’t very educational. It’s like leaving kids with a cliffhanger, except they don’t know that there’s a resolution. They’re left with “well, that was easy. Wait, why did we do that again?”

Students constructing their own analogies and making their own connections is when the true learning happens. The idea behind this activity is to craft an experience such that students are armed with observations and data they collect. Then, they use these data and observations to justify their explanation on what’s happening, creating one connection point. Further than that, though, is that they draw parallel connections between two activities that seemingly have nothing to do with each other at the surface. These connections allow students to visualize a phenomena that they previously had no visualization for, or, if they had one, a sorely inadequate one. And they draw these parallels by themselves. I cannot stress the importance of that!

Some tips on fostering quality classroom discussion

A crucial part of this activity was the classroom discussion. One could write a novel on how to do this, and I’d like to do a separate, more detailed post in the future, but here’s some bullet points for now:

  • Push, push, and PUSH students to elaborate on their predictions and explanations and to justify those with evidence.

    That’s an interesting idea. Would you explain more about what you mean?
    What exactly do you mean by [word/phrase]?
    What about your observations/evidence led you to conclude that conclusion?
    What is your evidence to support that?
  • Reinforce to students the difference between an explanation (or theory or model) and supporting evidence. Without evidence, explanations are just words, and as the denizens of Westeros like to remind us: Words are wind!
  • Summarize explanations that students give for the rest of the class. So, what you’re saying is [summary]. I just wanted to make sure I understood your idea. Does that sound right? But make sure you’re not inserting your own words or ideas!
  • Link student ideas together. If one student’s idea is related to the student that just spoke, call them out by name. So, Billy’s idea feeds right into Cindy’s because [reasons]. Better yet, ask the class how the two ideas are related!
  • Don’t let students say “pretty much what they said”: Ok, so you agree? Would you explain it in your own words? I want to hear what you think!
  • Do your best to acknowledge merit, no matter how small, in each student’s idea.
  • Praise students for their ideas, and make that praise specific.