Abstract

Faraday’s Law of induction is often stated as “a change in magnetic flux causes an electro-motive force (EMF)”; or, more cautiously, “a change in magnetic flux is associated with an EMF”. It is as well that the more cautious form exists, because the first “causes” form can be shown to be incompatible with the usual expression V = − ∂ t Φ , where V is EMF, ∂ t is a time derivative, and Φ is the magnetic flux.This is not, however, to deny the causality as reasonably inferred from experimental observation—it is the equation for Faraday’s Law of induction which does not represent the claimed cause-and-effect relationship. Unusually, in this induction scenario, the apparent experimental causality does not match up with that of the mathematical model. Here we investigate a selection of different approaches, trying to see how an explicitly causal mathematical equation, which attempts to encapsulate the experimental ideas of “a change in magnetic flux causes an EMF”, might arise. We see that although it is easy to find mathematical models where changes in magnetic flux or field have an effect on the electric current, the same is not true for the EMF.

Highlights

  • Michael Faraday was the first person to publish experimental results on electromagnetic induction, a phenomenon which is frequently described as a current being induced when the magnetic flux through a conducting coil is changed.The phenomenon is compactly described by a mathematical model relating the “electro-motive force” (EMF) V around a closed circuit to the temporal changes in magnetic flux Φ—the integral of the magnetic field over the surface delineated by that closure

  • To address this apparent deficiency, we derived a Faraday-like law based on the Maxwell Ampere curl equation, which allowed us to talk of induced changes in the EMF, but these were instead caused by spatial gradients in the magnetic flux, which is not quite what we had hoped

  • When the mathematical model was converted to refer instead to induced changes in an EMF-like quantity V L, we found that the model became incompatible with the desired causal interpretation, emphasizing that it is an insistence on using EMF

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Summary

Introduction

Michael Faraday was the first person to publish experimental results on electromagnetic induction, a phenomenon which is frequently described as a current being induced when the magnetic flux through a conducting coil is changed. Where we are likewise compelled to describe the spatial gradients of the electric field ∇ × E as a cause, and the resulting temporal changes in field B as its effect Somewhat disturbingly, this means we are unable to interpret our mathematical model of Faraday’s law of induction as our preferred causal statement: i.e. where changes in flux induce an EMF (and drive currents) [16,17,18]. While we would usually hope that these agree, it seems that in the case of Faraday’s law they do not, and in Section 2 we discuss why (or how) this can be It would be desirable for our mathematical description of induction to explicitly show how EMF or electrical current could be generated in a conductor, whether by the varying properties of the magnetic field, or by motion of the conductor within those fields. An alternative focus on the electrical currents generated by motion and or magnetic field variation has no such limitations

Experiment
Maxwell–Faraday Equation
Maxwell–Ampere Equation
Lorentz Force Law
A Simple Loop
U-Shaped Bar and Moving Rod
Constant Magnetic Field
Time Varying Magnetic Field
Electro-Motive Force
Discussion
Conclusions
Full Text
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