We were unsatisfied with our capacitor model. What makes the charges move? Why does this happen? So we made a sharp left turn into sticky tape and static electricity. We came up with experimental verification of why there are only two charges, and we even saw the electric force change based on distance. We'll get more mathematical tomorrow.
I had students answer some questions from the end of a PHeT lab on capacitors after the test today. The most interesting conversation came from the first picture above. How do places of "more than typical amount of charge" interact with other areas with "more than typical amount of charge"? With areas with "less than typical amount of charge"? We had no good model, but more than one student came up with more than one circuitous way to draw the forces on the bottom right negative spot like in the picture above. We're sure it's the right answer; we just don't like it.
We also figured out how capacitors add in series and in parallel from first principles today. It was quick and not a focus of the lesson; I don't think it's essential to the curriculum, but it's good practice of Kirchhoff's Laws.
We used the PHeT simulation to look at how capacitors act in circuits. We quickly saw relationships between charge and voltage for a given capacitor and between the physical qualities of the capacitor and its capacitance.
This is new time in the year to talk about capacitors for me. Usually I don't introduce capacitance until we already have a detailed model for the electric force. It seems to work well, but we'll have to loop back later to talk about dielectrics.
Today was mostly spent practicing our model of internal resistance and making our models for the ammeter and voltmeter clear. But we did spend a little time figuring out how capacitors act when completely uncharged and when they've been in the circuit for a long time. We also looked at what happened if we charged a capacitor with 3 D-cells versus 6 D-cells. It seems like a lot more energy is stored when a larger potential difference is across the two terminals.
We spent a lot of today using Kirchhoff's Laws (along with Ohm's Law) to make sense of circuits from the TIPERs book. The questions in TIPERs are hard and intriguing, and one student even came up with an alternative answer for a problem that I would never have come up with on my own.
Then, it was time to make sure our models for circuit elements were as precise as possible. We feel good about our model for resistance, but what about for batteries. Do they always have the same voltage, no matter what? So we tested any and all circuits we could, looking at just the voltage across the battery and the current through the battery. We had some pretty creative circuits to test this proposition.
I changed what I was going to do today due to student interest.
First, a little background. A student noticed that, if you add the reciprocals of the resistances of a parallel circuit and multiply that by the voltage of the battery, you get the current out of the battery. We also had a question with two resistors in parallel (48 Ω and 12 Ω) in series with another resistor (9.6 Ω). The two resistors in parallel shared the voltage equally with the one resistor, and the sum of the reciprocals of the two resistors (1/48 + 1/12) equaled the reciprocal of the sole resistor (1/9.6). It was as if the reciprocal gods were controlling circuits.
So, after doing some TIPERs today, we tried to figure out why this happened. I showed the proof for the equivalent resistance in series and in parallel, and so many a-ha moments happened. We then did some equivalent circuit problems, and we saw them as cool logic puzzles to solve.
The lab today was amazing. My AP Physics C students from last year set up these hidden circuits for each other. They had five or six resistors in them, and they tucked all the wires inside an overturned box so that only the resistor and the tips of the wires attached to the resistor poked out of holes in the box. They were difficult and students used a lot Kirchhoff's Law to figure out the underlying circuit structure when they could only measure voltage across and current through the resistors. I had them simplify the circuits so there were only three resistors. Those I kept, with the schematic diagrams they drew, until today.
I put them in groups of 5 groups of 4 around the 5 different circuits in the room. They together tried to figure out how the three resistors would hooked up in the circuit. Then I told them the four people in their group had to scatter to four different tables, and we did a second round. And then the third round.
And then the insight happened. Instead of just showing them the answer keys, we would make the answer keys. Those students who analyze the circuit in lab had to go to the front of the board and discuss how they think the circuits were wired. Those who never got to see that particular setup took it apart and, from the jumble of wires, tried to figure out how it was set up. Those students also used the ohmmeter function of the multimeter to find the resistance of the resistors in the circuit.
This was great. The students argued in the front of the room. I taught the students in the back how to use the ohmmeter and how to look at a tangle of wires and draw an ideal circuit.
The most discussion happened in the circuit where all the resistors had the same voltage drop but where the current in one resistor equaled the sum of the currents in the other two resistors.
At the beginning of class, we whiteboarded some problems, and I took pictures of them, but I'm not even including them. The lab after was better.
About half of my students were gone on a music field trip. What to do? Independent work time to use our model of electric circuits.
We then used what we knew to deal with circuits where there are two resistors in parallel in series with a single resistor. It's three equations and three unknowns fun!
Now that we have definitions of current and voltage, we can see how, when we vary the voltage across a circuit element, how the current in that circuit element changes. We tested a couple of different resistors and the long bulb from the CASTLE kit:
We learned how to use voltmeters and ammeters today. We wanted to quantify how to measure electric pressure difference and flow rate.
After measuring electric pressure difference in seven different circuits at many different points in every circuit, we whiteboarded our data with our physics goggles on. (By physics goggles, I mean without making a big deal about tiny differences and while looking for patterns we see in the data.) We seemed quickly to come to consensus about how the electric pressure goes up in the battery and then down in the lightbulbs.
We then changed our multimeters from voltmeter mode to ammeter mode. I made a big deal about this this year; after replacing seven fuses last year, I wanted to put a fear into using the ammeter. I might have given them too much fear. I'll take it, though. After whiteboarding the circuits from the ammeter, we saw another pattern. The flow stayed the same unless it split.
We'll see how current and voltage relate tomorrow.