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Sunday, July 17, 2016

Unit 2 Work Product - 2016 E&M Modeling Instruction Workshop

'*In every modeling workshop there seems to be that moment when your brain explodes in a sudden clarity of understanding. This is that unit for me. This was the area I was weakest in understanding and just went through the motions with my students. The sad thing is I thought I understood it well enough. I definitely did not. On with the show.

Day 4 began with Don doing a bit of a review on proportional reasoning using the equation we derived on Day 3. An important bit of language was added to our vocabulary. If the distance doubled between to charges what happens to the force? Rather than saying it quarters, or it goes down by a fourth he asked us to use the following phrase: It decreases (or increases) by a factor of #. Where the # gets replaced with an integer. So if the distance doubles, the force decreases by a factor of 4. If the distance triples, the force decreases by a factor of 9. Half the distance? The force increases by a factor of 4. He also used the graph from our lab to reiterate this relationship that was very helpful visually.

He then brought our attention back to the fact that the Coulomb's Law and the Universal Law of Gravitation were so similar. We have been using this concept of gravitational field strength in our class for some time now. But haven't really talked much about them. The gravitational field of the earth is what gives an object weight. It pulls roughly 10N for every 1kg. If I double the mass, I double the force. This happens with charged objects too. So in this way we can think of charged objects of having an electric field. Since our units for grav. field are N/kg an analogous idea for charges would be the N/C.  If I double the amount of charges the force increases by a factor of 2. Grav force can be calculated using the formula F=mg. Elec. force can be calculated using the equation F=qE. Since forces are a vector component, one of the variables on the other side of the equation must be a vector too. Mass can't be, and charges have pos. and neg. but it doesn't indicate direction of anything so we concluded that fields must have direction. Gravity fields are easy, they only pull one way so the earth's gravitational field points straight down. But we have different charges for electric fields so how will that work? Convention time! Evidently it was decided long ago that we will work with positive charges when determining the forces on charges. So the E. field points in the direction that a force would act on a positive charge due to the field.
So we came up with 2 rules about drawing electric field lines before we did some problems.
1. lines are drown in the direction of the force on a charge placed in the field.
2. closer spacing = stronger force.

As you move away from the center charge the lines get farther apart so force decreases.

If we have two long oppositely charged plates it forms a relatively constant electric field  in between the plates. We know this because the field lines would all be equally spaced and parallel to each other. (It breaks down around the edges though so stick to the middle!)

We then reviewed energy and the fact that energy can be:
1. Stored
2. Transferred
3. Conserved
And since we can't see energy we have to classify it by the way that we observe it. (Kinetic (motion), Grav. (height), elastic (stretch/compression), heat (temp)) So what about electrical? We discussed it using an eraser in the gravitational field. When you lift the eraser does the eraser itself have more energy? The discussion led to that the energy is stored in the field itself. I don't know how to feel about that. Stuff we can't see is stored in something that is not really an object that we made up to explain the idea of things affecting other things at a distance. I don't like when we just have to accept something at face value but I don't see a way to come up with this idea other than how we did it. Another thing to keep thinking about.

We introduced the idea of gravitational potential and electrical potential. It is sort of like field strength for energy. Field strength was the force per unit of mass/charge, potential is the energy per unit of mass/charge. I think this is a particularly difficult idea just because it is something that really never gets used in normal conversation other than with batteries. Let's be honest, very few people understand what potential means with batteries either. It is just a word that is used with very little context or understanding around it. We began talking about potential using contour maps and dealing with the grav. field. Potential can be seen as a hill. The steeper the hill the stronger the localized field is. The taller the hill, the more potential there is. It was easy to wrap our head around because most of us had seem something like this before. Electrical equipotentials were a bit more difficult. We have opposite charges now.

Grav energy = mgh           Elec energy = qEd
   Grav potential=energy/mass=gh           Elec potential=energy/charge=Ed

At this point we did "the lab". I did not realize at the time how important this lab was, but we came back to it the rest of the workshop. Super important for our understanding. We placed two iron bars in a baking dish with a bit of water in it. We then connected them to a DC power source and used our potential-o-meters to measure the electrical potential at about a billion places. (Took forever!) we then graphed it using to get a graph that looked just like a hill! 

We had noticed that the hill looked like it was constant in its steepness for the majority of it. We concluded that must be the field strength value. Some groups used different voltages on their power supplies and got different slopes. Some used less distance between the plates and got the same result. We realized that if we moved the bars closer the hill would get steeper because the top and bottom had to stay the same. This connected back to our contour maps. When equipotential lines are close the hill is steep!

We then used a phet simulator to see how this worked with point charges instead of plates. The hills did not seem to be constant around point charges. They looked more like this:

The valley is caused by a negative charge as all positive charges would roll downhill towards it. Note that the hills change steepness as they get away from the charge. This goes back to the inverse square between distance and force on field strength. Interestingly, the inverse square thing doesn't seem to apply to the charged plates. We talked about this and the charges push on each other and pull on each other and based on superposition the forces stay pretty much equal. As you get farther from one plate you get closer to the other plate so it all stays constant.

Anyway, things were going very smoothly until we hit problem 11.

This problem made my brain weak. At first glance it looks EXACTLY LIKE THE LAB WE DID!!!! So my lab group breezed through it. Then Don and Laura kept coming by and reminding us that they were not hooked up to a power supply in the problem.
Us: Yeah. We get it. (don't change anything because we are so amazingly right)
Them: But the power supply was keeping the potential constant.
Us: Yeah. Cool. No charges can leave or enter so.... We're good!
Them: Really?
Us: Yes?
Them: Really really?
Us: Look energy is increasing since the plates are being pulled away right? So the field strength should get smaller and the distance increases and so the potential stays the same. We. Are. Good! You are not going to get us with your Jedi Mind Tricks!
(They go away) Us in our group: Ok. Why do they keep asking us this. Are we doing something wrong? Potential has to be the same right? No charges can move! When plates move away we know the field gets smaller we saw it! So what could we be doing wrong?

Eventually we got back in a group and discussed this problem.
Laura brought up this idea. If a charge is very close to one plate, pretty much only the charge next to it is pushing it away. All of the others are pushing on an oblique angle to it. Their forces balance out and so the charge gets pushed straight out from the plate. As the charge moves away the force from charge 3 gets reduced at the inverse square rate but here is where my mind exploded. At this point all of the charges around it have a greater effect because the angle they push at is much closer to 90! If you imagine the plate to be a huge wall (from a single charge perspective), the area of charges increases as a square to the distance! Yes, the one charge is less but the others start to help out more and so the force ends up being constant! Holy crap this changes everything! As the plates in question 11 separate the FIELD STAYS THE SAME!!!!!!!!!  Energy goes up proportional to the distance moved. Since energy is equal to qEd and elec potential(V) is energy/q then Ee=qV. I can't believe we didn't see this before! If charge is not changing but energy is increasing, the potential CAN'T BE CONSTANT!!!!

Sorry for all the shouting. It was a very big moment for me in this unit.

Once we had this background set we were given these large blue cylinders that we were told were energy storage devices. (Later we learned they were called capacitors) A capacitor is basically two really big parallel plates rolled up into a smaller space but we could move the charges from one plate to another using a battery or a generator. (Note: This is not charging it! The device is still neutrally charged. Charges have just been moved from one plate to the other. We called this energized.)

There were various capacitors and we did a lab seeing how much energy was stored in each type. We found that as we increased the potential the energy increased as a square of the the change in the potential. The coefficient we labeled as 1/2 the capacitance. We then did a phet sim on capacitors to determine what things affect the capacitance. We also looked at using multiple capacitors. If we hooked them up in a straight line energy for each added capacitor went down! When we hooked them where one side was connected to one terminal of the battery and the other terminal to the other side (what we would later know as parallel) the graph of energy per capacitor added was linear! Crazy! As we thought about it it made some sense based on what we had done with the phet sim. Adding capacitors in parallel was like increasing the area of the plate which we saw had a direct relationship with energy. 2 capacitors = twice the area. For the one in series, the total potential was the same for all capacitors in a line so if there were 2 capacitors, each one only stored half the energy.