What does the conducting paper lab teach?

At my school we have a lab for electric fields/potentials where they measure potentials at different locations on a piece of conducting paper that has been attached to a power supply (the + lead and – lead of the power supply are placed at different points and the voltmeter + lead is anywhere on the paper and – lead is connected to the – lead of the power supply). We ask students to find equipotentials on the paper and then to plot what the electric field must look like by plotting vectors at all times perpendicular to the equipotentials. The strength of the field comes from the density of the equipotentials.

In principle this is a cool lab (and not as sadistic as the ones that electrify goldfish bowls trying to get the fish to line up on equipotentials to minimize the pain). However, for me, it has some problems. First, it’s not an electrostatic situation. We don’t have charged locations on the paper, so saying that it’s measuring potentials and fields is not correct. Instead it’s a very complex circuit with current flowing in all kinds of weird ways. Second the best analogy is with infinite lines of charge, instead of point charges. I find it hard to get students to understand that, mostly because they can see that it really doesn’t have infinite lines of charge sticking out of it.

I wanted to see how good that analogy was, so I set out to calculate the current flow through the paper. I first did some thinking about doing this while considering the paper to be a continuous resistor. Unfortunately I’ve only had success doing that sort of thing for very symmetric situations like a cone-shaped resistor (a popular homework problem that has problems of its own). So I decided to look into what it would take to model the paper as a grid of resistors. Here’s a sample of the model I built:

sample of the resistor grid model under consideration. The red arrow indicated the guessed direction of the current.

sample of the resistor grid model under consideration. The red arrow indicated the guessed direction of the current.

Ultimately I settled on a 30×30 grid as a compromise between good resolution and speed of calculation (everything below takes ~3 seconds to run on my Surface Pro 2).

Using an approach I’ve talked about before, I set out to figure out what equations I could use to solve for the unknowns (all the currents and all the voltages at the grid points). Here’s what I came up with:

  • For every node points: current coming down from above + current coming in from the left = current leaving going down+current leaving going right
  • For every node point: V[{i,j}]=V[{i+1,j}]+currentdown[{i,j}]*R (I set R to 1)
  • For every node point: V[{i,j}]=V[{i,j+1}]+currentright[{i,j}]*R

I had to do some careful work around the edges, and I had to set one location to the voltage of the power supply (I used 1 Volt) and another to zero. I also had to name the current through the battery as an unknown.

All together I have 2642 equations and 2641 unknowns. Mathematica is then happy to oblige. Here’s a plot that has a lot of information. Color is the potential at every point. The thick black lines are equipotentials. The arrows are the direction of current at each point (really a vector of both currentdown and currentright). The opacity of the arrows is the strength of the current.

+1 volt is at the red portion and ground is at the blue portion

+1 volt is at the red portion and ground is at the blue portion

Pretty cool, huh. The image is done with a single command in Mathematica called ListStreamDensityPlot. Awesome.

Ok, so how about the analogy? Here’s a comparison of the voltage along a line that connects the red and blue points. Blue is the results of this calculation. Red is the best fit for the potential due to a positively charged infinite line of charge coming out of the paper at the red point and a negative one at the blue point. Green is the best fit for the potential of point charges at red and blue.

Potential along line between the connection points. Blue is this calculation, Red is the best fit for infinite lines of charge. Green is for point charges

Potential along line between the connection points. Blue is this calculation, Red is the best fit for infinite lines of charge. Green is for point charges

By my eye the red fit is much better, at least in the interior. I actually expected some discrepancies near the border of the grid anyways.

So, here’s my question: is this a good lab to do? I’m not convinced. Yes, there’s a decent analogy with something the students could do as homework, but the physics is very different (current is flowing in this case, for example).

Your thoughts? Some starters for you:

  1. I’m in this class and I thought it was a great lab. Here’s why . . .
  2. I’m in this class and I’m supposed to be studying for my oral exam right now, not reading this!
  3. Wait, what did you say about goldfish?!
  4. Can you give more details about what you did around the edges?
  5. Can you give us your code?
  6. I do this lab and think it’s great. Here’s how I handle your objections . . .
  7. I hate this lab and am glad I have this to point to in the future, thanks!
  8. Can you do more interesting designs like we do in lab? Bars of connected points all at the + voltage, circles, etc?
  9. How did you define your zero of potential for the infinite lines of charge? Doesn’t the log function get you into trouble?
  10. I’ve talked about this at length and I can’t believe you didn’t link to me, jerk.
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About Andy "SuperFly" Rundquist

Professor of physics at Hamline University in St. Paul, MN
This entry was posted in fun, mathematica, physics. Bookmark the permalink.

5 Responses to What does the conducting paper lab teach?

  1. Juan J. Patron-Diaz says:

    Now Im curious as to what we actually did during that lab?

  2. Hailey K. says:

    I am in this class and should be studying for my oral exam right now, but thinking about this is still somewhat relevant. I think that this lab did help illustrate the idea of an electric field going from positive to negative. I don’t fully understand the complexity behind your objection, but I do get that the lab was supposed to be about fields and instead the example was through a circuit. However, I think labs success should take into consideration learning outcomes. I wouldn’t want to learn anything incorrectly, and maybe this lab has given me some false sense of how to create a field, but I did understand fields better. The idea that they went from positive to negative was made very clear as well as the fact that different point charges and shapes will create different fields. I agree that an infinite line scenario may have been more useful and correct, but practically more difficult. Compared to labs with carts and ramps I definitely liked this lab more and still feel I absorbed something of worth. What are some other ideas/ways that this concept could be taught? The goldfish experiment doesn’t seem very nice or practical either.

  3. Kevin C. says:

    I’m in this class and I quite liked this lab. I can understand your reservations about the lab representing something that is not actually happening, but I found it to be a useful tool for learning the concepts. The potential problem of the lab creating a cognitive dissonance with your more keen students in my mind becomes a good thing as those students will likely seek you out with questions. That sounds like a fun conversation to have with interested students!

  4. Asim says:

    I would be very interested in knowing the conductivity of the material used, and the thickness of the actual conducting material spread on the sheet. Thank you.

    • Andy "SuperFly" Rundquist says:

      I don’t have answers for either of those except to say that the conductivity wouldn’t affect the image as long as it was isotropic

      On Tue, Jan 6, 2015 at 12:18 PM, SuperFly Physics wrote:

      >

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