What is the difference between motional EMF = $-vBL$ , and Faraday's law of induction $\displaystyle\mathcal{E} = \left|\frac{d\Phi_B}{dt}\right|$? Aren't they the same? What is the relation of Lorentz force to motional EMF?
Answer
Faraday's law $\mathcal{E}=-d\Phi/dt$ can be used in a variety of situations, including ones where the phrase "motional EMF" is appropriate.
Your particular expression $-vBL$ is applicable only for a very particular situation. Probably a sliding bar, which is part of a circuit, in a uniform magnetic field. That expression can be derived using Faraday's law, and is a one- or two-liner if you go through it. I believe you can derive it using other methods ($\vec{F}=q\vec{v}\times\vec{B}$ and such), but Faraday's law is applicable here too, and in so many other situations where that force law would give misleading answers.
So I suppose the phrase motional EMF is used when there is physical movement of a conductor. The term Faraday's law is typically used to indicate the method one uses to calculate what the EMF is.
To address your Lorentz force law question more explicitly: Faraday's law is used especially in situations where $F=q(\vec{E} + \vec{v}\times\vec{B})$ might yield a misleading answer since an induced electric field causes by a changing magnetic field causes the force, rather than a magnetic force as one might expect. (Well, that's the usual interpretation. SR grumble grumble.) You don't run into this situation with the sliding bar example, but if you have a stationary conducting loop immersed in a changing magnetic field, one might ignore the electric field in the Lorentz force law since you're not actively creating such a field. But actually there is an electric field; Faraday's law tells you what the path integral of that electric field is, which is useful.
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