Learning difficulties among students when applying Amp\`ere-Maxwell's law and its implications for teaching

We investigate learning difficulties among second-year students on electromagnetism courses when they apply Amp\`ere-Maxwell's law. Using phenomenography, we analysed written answers from 65 undergraduate physics students to four questions on Amp\`ere's and Amp\`ere-Maxwell's laws. We complemented our research by interviewing twelve students. To design the questionnaire, we ran an epistemological analysis of classical electromagnetism which helped us to identify a set of key essential concepts to understand this theory and guided the definition of learning objectives and to draw up the questions. The results revealed that the students found it hard to recognise the validity framework from Amp\`ere's law and to apply Amp\`ere-Maxwell's law. They face particular difficulties to recognise the appearance of the displacement current and the relationship between the circulation of the magnetic field and an electric field that is variable over time.