Lenz’s Law Activity

This is the second year I’ve taught AP Physics C: E&M, and we’re getting to the final leg of the course: electromagnetism. I was preparing the unit’s opening activities when I came up against the struggles of Lenz’s Law. One pesky negative sign within Faraday’s Law creates more trouble than just about anything in this class.

At this point in the year, my students have studied electrostatics, including potential and Gauss’ Law, RC circuits, and magnetostatics, including Ampere’s Law.

I’ve never really understood how to use Lenz’s Law to consistently predict the correct direction of an induced current. I was trying to figure it out again a few days ago, and it ended with some choice 4-letter words and my head on my desk.

But I’ve got it now! And I’ve come up with an activity that will help students understand it as well all thanks to the HTML5 version of PhET’s Faraday’s Law simulation.

The HTML5 version is particularly good because of the option to display the magnetic field lines coming from the magnet. This is the key to the entire activity as it’s what allows students to reason through how the magnetic flux through the wire loop is changing. It also being an HTML5 sim allows it to run on phones and tablets. Also, it’s FREE.

Faraday’s Law First

I had students first explore Faraday’s Law through the old java version of the sim. This one allows you to vary the size of the loops, which the HTML5 version doesn’t. I ask students simply to play with the sim and figure out the conditions required for current to be generated in the wire as well as to list the factors that affect the magnitude and direction of the current. Students don’t have much trouble figuring all of this out.

We then have a quick discussion on how to predict the direction of the induced current, and this is where things get sticky. There are lots of patchwork theories that arise that are basically a list of if-then statements involving the direction of motion of the north or south end of the magnet relative to the loop.

This is ultimately unsatisfying because a list of if-then’s is not a good scientific model, especially situations that aren’t “moving a bar magnet near a wire loop.” I then take a few minutes to review flux, which they’d seen from electrostatics, and then introduce magnetic flux as basically the same thing, but with magnetic fields.

I conclude the discussion by emphasizing that looking at this from a “how is the flux changing” perspective is more generalized than “which end of the magnet is moving how” perspective and, thus, will result in a much more generalized and useful model.

Data Collection

The goal of the activity is to collect “data” by drawing before-and-after pictures of the magnet moving into and out of the loop. Students use each set before-and-after drawings to figure out the change in magnetic flux due to the motion of the magnet, the voltage reading (whether it’s positive or negative) to determine the direction of the current, and then the right-hand rule (which they know from the previous unit) to figure out the field created by the wire.

Here’s one example of moving the north end of the magnet towards the loop. Having the “field lines” box checked is essential.

Before:

lenz1

After:

lenz2

In the “before” snapshot, students draw an arrow* indicating the direction of the flux through the loop. In the “after” snapshot, students draw another arrow showing the direction of the flux, longer if the flux increased and shorter if it decreased, then a dashed arrow showing the change in the flux. They also use the voltage (negative voltage = flow from positive terminal to negative terminal, positive = opposite) to determine the direction of the current, which they draw, and then they draw the direction of the field in the wire due to that current.

To figure out if the flux increases or decreases, they use the density of the field lines in the loop. More density = greater field = greater flux. The direction of the change is figured out by looking at the direction that they had to stretch or compress their initial flux arrow to turn it into the final flux arrow. For example, the first picture has a flux arrow pointing to the left, and the after has one longer, but still pointing to the left. If you wanted to make the first one look like the second, you’d have to stretch it to the left. Therefore, the change is to the left.

Students look at 4 scenarios, which require 2 drawings each:

  • The north side of the magnet moves from to the right of the loop to the front edge being inside the loop (pictured above).
  • The far end of the south side of the magnet moves from the center of the loop to the left of the loop (pictured below).
  • The far end of the south side of the magnet moves from the left of the loop to the center of the loop.
  • The far end of the north side of the magnet moves from the center of the loop to the right of the loop.

Basically they pull the magnet into and out of the loop from the right, then reverse and pull it into and out of the loop from the right.

lenz3

lenz4

*Since flux isn’t a vector, I’ve been careful not to call them vectors. Though, what we’re kind of doing is vector subtraction to figure out the direction of the change. I’m not quite sure why we can treat it like a vector and it still work out that way. Oh well.

Data Summary

Once they’re done, they’ve got 8 drawings with all kinds of arrows, so it’s kind of a mess. The goal is to see a pattern that they can use to predict the direction of the induced current based on the change in flux. When they’re eventually figuring this out on their own, I want them to think about it like this:

  1. Figure out the direction of the change in flux.
  2. Figure out the direction of the field that would oppose that change.
  3. Use the right-hand-rule to figure out the current required to generate that field.

To facilitate that, I had my students make a summary page that filtered out most of the data so that they could easily see the pattern. I put this up on the board and told them to fill in the rest from their data once they finished.

14A571DA-48A7-401F-BC20-F9FDCECF0E96

The conclusion to this was to figure out the pattern between the direction of the change in flux and the direction of the induced magnetic field. If done correctly, all of them should be opposite of one another, which students easily saw.

Here is an example of some student work.

1BBC7FCB-63D5-4694-AF45-EE08AC3DB4F0

19B3A3AB-280C-43CF-8C28-CD9A234EA441

A few other things

The devil is in the details for this activity, so I walked through the first set of drawings with everyone to make sure they were doing everything correctly. I told them that they needed to draw the same number of field lines for each snapshot (I chose 6, the simulation draws 8), and that when drawing them, they needed to make sure and basically draw them as they’re drawn on the simulation. Doing this right is key to being able to use the density of field lines to deduce the magnitude of the field/flux within the loop.

My students also have lots of practice in figuring out the direction of change by looking at before and after vectors. Even if they hadn’t, explaining it as I did above (by looking at the direction required to stretch or compress one vector to the other), I don’t think it would have been too difficult for them to figure out.

I also made sure to check on each group frequently to make sure they were doing everything correctly because, again, getting everything correct is key to seeing the pattern at the end.

If you have any suggestions for improving this, please let me know! If you use this (or something similar) and would like to share your experience with it, please do so as well! And if you need help or clarification with anything here, leave a comment, and I’ll do my best to help out.

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!

 

 

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.

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!