Finding Derivatives from an Equipotential Graph

We examine how upper-division physics students find derivatives from an equipotential graph in an individual interview setting. We focus specifically on identifying the different kinds of behavior that students engage in when finding a derivative from an equipotential graph, and also on the representational elements that students use or introduce during their work. We find that the students were able to find the derivative successfully using a ratio-of-small-changes approach. Students engaged in behavior like sketching straight lines or arrows on the given graph as a way to choose points for calculating a ratio and to keep one variable constant for the derivative. We also saw students make sense of the equipotential graph and reinterpret the derivative they found using those representations using two other graphical representations: cross-sections and three-dimensional plastic surfaces. We find these results encouraging because the students had studied equipotential graphs and derivatives as ratios-of-small-changes as part of their junior-level electrostatics, suggesting that such a course is effective at helping students develop representational fluency for working with multivariable derivatives.