Challenges in applying the magnetic field circulation in Amp\`ere-Maxwell's law

We investigate how students in introductory electromagnetism courses apply the idea behind the magnetic field circulation in the context of the Amp\`ere-Maxwell law. For this purpose, we designed a metacognitive pencil-and-paper question, presented it to 65 students, and analysed their responses using phenomenography. We complemented our research by conducting interviews with 12 students. The results show that students tend to think that they can only use the magnetic field circulation to calculate magnetic fields if the curve used is symmetric, and that they do not take into account the shape of the magnetic field lines when applying it. We also find that some students believe that a variable electric field must cross the entire Amperian curve in order to apply Amp\`ere-Maxwell's law and find the magnetic field.