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!

 

 

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