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 , where 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 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 the “flux rule” is not a physical law but only a calculation shortcut that does not always yield the correct predictions. This 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.
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The name “electromotive force” is misleading, because, as we know, the electromotive force is not a force: it has the dimensions of an electric potential, and it is measured in volts. Since a more suitable name has not been invented, we shall keep on using the same name. It is worth noting that, today's use of some historical names appears to have no justification. For instance, we find that the magnetic field
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The above discussion reminds us of another question. Let us consider Maxwell's equation: It is easy to find statements according to which this equation shows that a time-varying magnetic field causes an electric field (and symmetrical statements for the equation of the curl of the magnetic field). See, for instance, Ref. 20, where statements of this kind are considered to be conceptual misconceptions that one should avoid in teaching. The equation states only a relation between the fields as the charges produce them: see also Ref. 21. This is well illustrated by the equations that yield the fields produced by a point charge in arbitrary motion. The electric and magnetic fields are given by equations that depend on the charge's velocity and acceleration. These equations are independently deduced one from the other (see, for instance, Ref. 15, pp. 870–879). Only ex-post, do we find that the two fields are related by the equation where is the unit vector pointing from the retarded position of the point charge towards the point at which the field is calculated. In the same spirit, let us emphasize that the presentation of Maxwell's equations in integral form deals a mortal blow to the intrinsic local nature of Maxwell's theory, and it transforms the production of electromagnetic waves into a profound mystery. These equations connect what happens on a closed line at time t to what happens simultaneously on an arbitrary surface having the line as a contour. This habit is widespread in textbooks for high school in Italy but not—for instance—in textbooks published in the United States.
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The applicability of this proposal to high schools depends on the mathematical background of the students. It is likely applicable in Italy's scientific Lyceums, where mathematical knowledge is sufficient to treat electromagnetic induction as here suggested.
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2023
Author(s)
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