Did this activity near end of the Paradigm course. I took about 10 minutes of class to “set it up”. First I pointed out the puzzle that V is a scalar field, E a vector field, yet despite the appearance that vectors carry “3 times the information”, we know that E and V share the same physical information content. How can this be? A student pointed out that the gradient carries information about V nearby a point, that's where the extra information is hidden! This led to writing V(x,y,z) on the board, then underneath E = - (dV/dx $\hat x$ + dV/dy $\hat y$ + dV/dz $\hat z$). Spend a couple minutes talking about how this means that the components of E cannot be “random”, they are all coupled/connected in important but subtle ways. Set up the “murder mystery”: V is the murderer, but all we have are eyewitness reports, namely the 3 components of E. Can we reconstruct the murderer from this information? The activity is to interrogate (integrate ) the components of E and see if you can come up with a single, consistent, potential function.

My groups spent  35 min on this 3 page activity, and none was completely done. The first example (which looks like $-y \hat x + x \hat y$ stumped them all. Of course, this field has non-zero curl, it cannot be derived from a scalar potential, that's the point, but they couldn't bring themselves to believe this, usually tasks have solutions! The inconsistency of their “x” and “y” integrations was baffling everyone. Several groups wanted to add “xy” (from x integration) to ”-yx” (from y integration) and get 0 for the voltage. Working with individual groups we managed to get everyone past this. (One group even insisted that yx-xy was NOT zero, that since the terms “came from different places” they couldn't cancel. Interesting…)

One group noticed curl(E) is nonzero for this field, and were a little angry/bothered by the activity, since they “knew” it wasn't possible, and felt we were forcing them in a pointless direction by trying to find a V that doesn't exist.

For part 2, several groups integrated $\hat r$ and $\hat z$ pieces to get two identical voltage formulas, and then added these to get an extra factor of 2. (I pointed out that if one witness says the murderer is 6 feet tall, and a second witness says the murderer is 6 feet tall, do we conclude that the murderer must be 12 feet tall? This helped one group a lot!) One group was puzzled by the physical significance of this potential, they didn't realize that $1/\sqrt(r^2+z^2)$ in cylindrical coordinates is $1/r$ in spherical, the potential of a point charge.

Most groups sketched the first one, but fewer got to the second, and almost none completed the third activity - another 10 minutes would have been helpful here. Alternatively, if I did this again, I might have different groups start on different problems, so in the “student summary” at the end, each group would hear from other groups about what they learned about the other problems.

In the end, with a few minutes to go, I pulled everyone together and talked through the 3 cases, discussed what the graphs look like, which ones have curl, that we are requiring consistency, that not any random E field is consistent with the existence of a scalar potential… I pointed out that $\nabla \times \Vec E=0$ and $\Vec E = -\nabla V$ are two sides of a coin: that you can PROVE either one arises from the other, and thus that Maxwell tells us that the potential function exists. So, very high level of activity and engagement today, not so much forward progress as I had anticipated, but lots of questions and thinking going on.