Faraday's Law and Magnetic Induction: cause and effect, experiment and theory

Faraday's Law of induction is often stated as "a change in magnetic flux causes an 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 is incompatible with the usual expression $V = - \partial_t \Phi$. 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. Here I investigate a selection of different approaches, trying to see how an explicitly causal mathematical equation, which attempts to encapsulate the "a change in magnetic flux causes ..." idea, might arise.