In the Faraday-Neumann formulation the minus sign is a convention about how to orient relative to the normal that defines . Strip away the formalism, and what remains is a phenomenological statement so simple it deserves its own name: Lenz’s law.
Lenz's law
The current induced in a closed circuit always flows in the direction that opposes the change in magnetic flux that generated it.
The practical advantage is enormous: Lenz is Faraday-Neumann “read from outside”, it settles the direction of the current without forcing us to impose orientations beforehand.
At bottom, this is energy conservation translated into electromagnetism. If the induced current flowed the opposite way, it would reinforce the change that generates it: the flux would grow even more, the current would increase in turn, and we would have built a machine that amplifies energy out of nothing. To avoid this embarrassment, nature chooses the direction that brakes the change (Mazzoldi, Nigro and Voci 2008). See also Riferimenti bibliografici.
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Topics: Electromagnetic induction Concepts: Lenz’s law · Faraday-Neumann-Lenz law
Related exercises: Worked exercise — loop entering and leaving a field · Problem — Loop entering a field (qualitative) · Problem — Induced EMF over time (graph)