Students' conceptual understanding of electric flux and magnetic circulation

Electricity and magnetism are closely related phenomena with a well-known symmetry found in Maxwell equations. An essential part of any electricity and magnetism course includes the analysis of different field source distributions through Gauss's and Ampere's laws to compute and interpret different physical quantities, such as electric flux, electric and magnetic field, or magnetic circulation. Still, some students have difficulties with these calculations or, in some cases, identifying the differences between those quantities. We present this article to explore and compare the challenges that students experience when asked to compute the electric flux (surface integral of the electric field) or the magnetic circulation (line integral of the magnetic field) in a nonsymmetric field-source distribution with two opposite field sources inside a Gaussian spherical surface or Amperian circular trajectory. The sample consisted of 322 engineering students finishing an electricity and magnetism course. They were presented with two parallel problems. Half answered one in the electricity context and the other in the magnetism context. After a phenomenographic analysis, our results showed that the students' conceptual difficulties in both contexts can be grouped into the same categories but are not contextually parallel, as has happened when analyzing other electricity and magnetism concepts. Our results also suggest that the magnetic circulation concept is far more unfamiliar to students than the electric flux. We propose several factors that could explain this finding and suggest teaching to address the conceptual difficulties identified in our analysis.