From data to structure: Construction, breakdown, and reconstruction of an empirical representation in a black-box RC circuit

Abstract

In introductory physics laboratories, a central instructional goal is to help students construct, evaluate, and revise mathematical representations from experimental data rather than apply given formulas. We present a guided data-driven activity in which an RC circuit is treated as a black box defined through observable input-output relations. Experimental and graphical procedures are provided, while standard RC circuit theory is initially withheld. Students first examine discharging data obtained under different nominal values of R and C. They rescale time using T=t/RC and compare the normalized voltage v=V/E across circuit conditions, bringing the discharging data closer to a common trajectory. Graphical linearization through a plot of lnv versus T provides a basis for proposing an empirical representation and allows RC to be interpreted as a time scale rather than being assigned that meaning a priori. When the representation proposed for discharging is tested with charging data, logarithmic linearity is lost despite the continued organization of the data by T. Replacing v with the distance from the steady state, 1-v, restores linearity, and the resulting linear relation provides a basis for proposing a revised representation for charging. The activity thus uses a familiar physical system to make representation construction, breakdown, and reconstruction experimentally explicit. It also emphasizes that an empirical representation must be accompanied by a specified domain of validity and that standard circuit theory can be introduced a posteriori to interpret the physical meanings that emerge from representational constraints.