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Comment from S. Pollock (CU Boulder, visiting OSU and teaching Paradigm “Vector Fields), Nov 2009:

For me, this activity got split over two class periods, which worked out well, because I was able to “pull the group together” a little bit in the middle. Had I not done this, I think some all-group triage in the middle would have been helpful: there is a lot of wheel-spinning that goes on in this activity. I think it's very productive, they need to wrestle with many of these ideas, but 50 minutes without any collective guidance seems too short a time to see many groups get to completion…

I set up the problem (showing a hoop, telling them we are given Q, R, and T [period]) and asking them to find the A field everywhere in space. I had up on the board the formula for the vector potential as a triple integral over space of the current density (numerator) divided by “curly R” (denominator), which I wrote as |r-r'| We had already talked through the *idea* of this integral, and I chose to discuss symmetry arguments before letting them go (so, the class had collectively decided the DIRECTION that A should point everywhere, and that it should be independent of phi, so they could safely PICK phi=0 if they wanted.) This was a decision I made - clearly one could let them work this out on their own, and I am not sure if I took away some valuable creative thinking from them. As it was, they still had plenty of things to think about and work on, though - so if this was added to the mix, the timing for the activity would likely need to expand (or be run differently). The other “gift” I provided was a discussion before the started about how the formula for A should be re-written if you have a current sheet (replacing J(r') dtau' with K(r') d(area)', or if you have a line current (I(r') dl'). This was again something that they might have thought about/worked on themselves, but frankly I didn't feel this group was well equipped to make that leap (we have not talked about delta functions in 3-d yet, nor have we spent much time on I vs K vs J, the different current densities). So here again, I talked it through (with feedback, questions, and comments from the class) collectively BEFORE setting them off. I didn't tell them which to use for this activity, but of course this discussion zoomed them into the “I dl'” form much more quickly.

So what did they struggle with? A lot of time (a surprising amount of time) went into figuring out “I = Q/T”! For some groups this took over 10 minutes. More time went into realizing I(vector) points in the phi' direction, and wrestling with “is it phi or phi'”? Time went into remembering or re-deriving the denominator, the “law of cosines in cylindrical coordinates” formula for 1/|r-r'|. Despite the explicit conversation about symmetry, there was good discussion and argument about whether/why they could set phi=0 in the denominator. In the “first half” of this activity, only one group, at the very end (about 25 minutes), realized they could and SHOULD write phi'(hat) in terms of Cartesian vectors, i.e. that we can work in cylindrical coordinates, but our basis vectors need to be Cartesian to deal with this integral.

The next class period, I started with where I thought most had gotten to (so, reviewed the I=Q/T idea, set up the integral formally, getting the a form involving an integral of phihat' dl' in the numerator, had the class collectively talk about this idea of finding the x and y components of phihat', and then set them off. I thought it would be 5 minutes to “finish”, but it took another 25! Several groups wrestled AGAIN with the law of cosines denominator. (A weekend had passed between classes) The groups all struggled mightily with the integral of the x component (which vanishes by a rather subtle math argument). They knew (mostly) that it SHOULD vanish, but we pushed them to convince themselves, and this took a lot of the time. I pointed out that the remaining non-zero integral was elliptic, and suggested that they then take some limits: what is A at the origin? On the zaxis? Far away on the x-axis? The “origin” question was largely worked out, but the zaxis took too much time, and essentially nobody got (in the 25 more minutes I had allotted) to the large-x limit issue.

Summary - an excellent opportunity for students to review ideas about “curly R” (even though we don't call it that), how to integrate over a vector, symmetry arguments, converting between cylindrical and cartesian, the nature of current densities - there's really a lot here.