Electromagnetic induction: how the "flux rule'' has superseded Maxwell's general law

As documented by textbooks, the teaching of electromagnetic induction in university and high school courses is primarily based on what Feynman labeled as the ``flux rule'', downgrading it from the status of physical law. However, Maxwell derived a ``general law of electromagnetic induction'' in which the vector potential plays a fundamental role. A modern reformulation of Maxwell's law can be easily obtained by defining the induced electromotive force as $\oint_l (\vec E +\vec v_c\times \vec B)\cdot\vec{dl}$, where $\vec v_c$ is the velocity of the positive charges which, by convention, are the current carriers. Maxwell did not possess a model for the electric current. Therefore, in his law, he took $\vec v_c$ to be the velocity of the circuit element containing the charges. This paper aims to show that the modern reformulation of Maxwell's law governs electromagnetic induction and that the ``flux rule'' is not a physical law but only a calculation shortcut that does not always yield the correct predictions. The paper also tries to understand why Maxwell's law has been ignored, and how the ``flux rule'' has taken root. Finally, a section is dedicated to teaching this modern reformulation of Maxwell's law in high schools and elementary physics courses.