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Faraday’s law of induction and Faraday’s flux rule are closely related and often used interchangeably, but careful treatments (especially Wikipedia and advanced discussions) distinguish two statements under the broader heading of Faraday’s law.
Core Distinction
In the literature, “Faraday’s law” commonly refers to two related but technically distinct ideas:

Maxwell–Faraday equation (the differential/field form of Faraday’s law of induction)
A time-varying magnetic field is always accompanied by a circulating (non-conservative) electric field:∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}In integral form:∮∂ΣE⋅dl=−∬Σ∂B∂t⋅dA\oint_{\partial\Sigma} \mathbf{E} \cdot d\mathbf{l} = -\iint_{\Sigma} \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}This is a fundamental field equation. It holds whether or not a physical circuit is present.
Faraday’s flux rule (also called the Faraday–Lenz law or, frequently, Faraday’s law of induction itself)
The electromotive force (EMF) around a closed conducting loop equals the negative rate of change of magnetic flux through the loop:E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}where ΦB=∬Σ(t)B⋅dA \Phi_B = \iint_{\Sigma(t)} \mathbf{B} \cdot d\mathbf{A} .
For a coil of N N turns: E=−NdΦBdt \mathcal{E} = -N \frac{d\Phi_B}{dt} .

The flux rule is the practical, circuit-oriented statement that engineers and most introductory physics courses use. It can be derived from the Maxwell–Faraday equation (plus the Lorentz force) when applied to a thin conducting loop.
How the Flux Rule Covers Two Mechanisms
The flux rule unifies two physically distinct processes under one mathematical expression:

Transformer EMF: Stationary circuit, changing B \mathbf{B} . The induced electric field from ∇×E=−∂B/∂t \nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t drives the current.
Motional EMF: Circuit (or parts of it) moving in a magnetic field. The force on charges is the magnetic part of the Lorentz force q(v×B) q(\mathbf{v} \times \mathbf{B}) .

Feynman emphasized that the flux rule works for both cases, yet the underlying explanations are completely different: induced E \mathbf{E} versus v×B \mathbf{v} \times \mathbf{B} .
Comparison Table

Similarities

Both ultimately express the same physical phenomenon of electromagnetic induction.
For a stationary circuit the flux rule is a direct consequence of the Maxwell–Faraday equation.
Both incorporate Lenz’s law (the negative sign / opposition to the change).
Both are essential to the operation of generators, transformers, inductors, and motors.

Important Caveats (Why the Distinction Matters)
The flux rule is not universally valid for arbitrary closed paths. Classic counter-examples include:

The Faraday disk (homopolar generator): constant flux yet a measurable EMF appears.
Certain rotating-plate or sliding-contact arrangements where flux appears to change but the actual EMF is nearly zero (or vice versa).

In such cases one must return to the more fundamental combination of the Maxwell–Faraday equation and the Lorentz force law, ensuring that the path used for the line integral moves with the conducting material.
In everyday usage, especially in introductory physics and engineering, “Faraday’s law of induction” and “Faraday’s flux rule” are treated as synonyms and both written E=−dΦB/dt \mathcal{E} = -d\Phi_B/dt . The finer distinction becomes important mainly when discussing foundations, paradoxes, or the relationship to Maxwell’s equations.

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