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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?

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The latter is a generalization from a simplified situation to which the former applies. – dmckee Mar 30 '14 at 2:17
Maybe irrelevant, but I would make one comment: They are two different formulations of the same thing, but they must be same, mustn't they? This was the initiation of a thought process which led Einstein to Special Relativity Source : Griffith's Electrodynamics (Pg 303) – Cheeku Mar 30 '14 at 3:22
@dmckee I have no idea what you mean... my understanding is limited. – XCIX Mar 30 '14 at 21:02
The $vBL$ form requires a very special setup to be correct. The time-derivative-of-flux form (which is true for a huge class of cases) can be found from the velocity form in that one simple case. That doesn't prove that it is true in general, but it is about as good as you can do without vector calculus. – dmckee Mar 30 '14 at 21:23
What about passing a wire through a magnetic field? – XCIX Apr 1 '14 at 8:09

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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Moving a wire pass a magnetic field is applicable? – XCIX Apr 1 '14 at 8:02
its seems that electric motors follow the same equation. – XCIX Apr 1 '14 at 8:09

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