Faraday's law and motional emf

Question

My understanding of motional emf is that one of the ways it is created is by moving a conductor (moving such that its orientation doesn't change) in a uniform magnetic field (non changing). An emf is produced due to the segregation of charges due to the Lorentz force experienced by the charges while moving in a magnetic field. Is that correct?

If that is the case, how do you reconcile this with Faraday's law, as Faraday's law requires change of flux and here flux is not changing?

Of course, if you are moving a conductor in a field such that flux is changing (like changing the orientation of conductor), EMF is induced and that can be given by Faraday's law.

But in this case, flux is not changing but emf is still being produced. How to explain that?

Answer 1

I'd like to take your opening comment as read, so far as the reason why EMF is produced in a conductor moving in a uniform magnetic field, and address the second part of your question, how to reconcile it with Faraday's Law, which as you state requires a change of flux to produce EMF in the given circumstances.

Faraday invented, I think, the homopolar generator (see picture at this site: https://en.wikipedia.org/wiki/Homopolar_generator), which certainly produces EMF as well as measurable current. My perception that there could be an assumption in your question of what constitutes a "uniform magnetic field" caused to me to check, for the first time in about 45 years (yes, I am "old"), the specifications of Faraday's original use of the Homopolar generator as he was first formulating "his"laws.

To address aspects of your question in the hope that it leads you to the answer you want, look at the picture of the homopolar generator. To me, it appears self-evident that the magnetic field at any part of the conductive disk is "uniform" (as in "unchanging"), therefore the logic, albeit perhaps simplistic, ought to indicate that according to your iteration of Faraday's Law, that his homopolar generator should not have produced any EMF due to the instantaneous constancy of magnetic field at any point on the conducting disk - except it does!

Obviously, Faraday got it right and a century and a half of brilliant people concur with it. I am almost wanting, as a result of your question, to discuss the semantics associated with "uniform magnetic field".

Of course, the other way of addressing your question is to state that there is no such thing as a "uniform" magnetic field anyway. Any measuring instrument for EMF would draw a tiny current from the moving conductor, so right away one has a magnetic field around the conductor, and as it is moving, the resultant dynamic distortion of the magnetic field means that it is no longer uniform.

And surely, if one is not measuring the EMF across the moving conductor, and the magnetic field is truly uniform, how would one know (or care) if the wire had EMF or not?.

Answer 2

As user8736288 pointed out, the elementary area is $dl\times dx$ for a segment $dl$ displacing by $dx$.

I just want to add that Faraday's Law is stated in different forms (what IS Faraday's Law may be different for different people)

The form you mentioned maybe called the loop form of Faraday's Law.

The "Sweeping form" of Faraday's Law states that motional EMF = $-$ rate at which magnetic flux is swept by a segment $s$.

It doesn't require $s$ to form a closed loop. It's easy to derive the loop form using the "sweeping form" + Maxwell-Faraday's Law (a separate thing!)

However, the loop form only works when there's a loop (you can also make an imaginary loop, it would work the same). When there's no loop, it may be preferable to use Lorentz force law (always true) or this "sweeping form" I mentioned.

You can checkout this post I wrote, if you'd like to learn more: https://physicintuited.wordpress.com/2021/05/07/why-faradays-law-is-weird/

Hope it helps!

Answer 3

$$\varepsilon =\int (\mathbf{v} \times \mathbf{B}) \cdot \mathrm{d}\mathbf{l}.$$ Now, if we use $(\mathbf{a} \times \mathbf {b} ) \cdot \mathbf{c} =(\mathbf{c} \times \mathbf{a} ) \cdot \mathbf{b}$, we can write: $$\varepsilon= \frac{\mathrm{d}}{\mathrm{d}t} \int \mathbf{B}\cdot \mathrm{d}\mathbf{S}$$ where $\mathrm{d}\mathbf{S}= \mathrm{d}\mathbf{l}\times \mathrm{d}\mathbf{x}$.

So the elementary area to consider for the flux is the parallelogram formed by $\mathrm{d}\mathbf{x}$ (along the movement) and $\mathrm{d}\mathbf{l}$ (along the wire).

Answer 4

You're correct that motion of a conductor through a uniform magnetic field segregates the charges due to the Lorentz force. See also the Hall effect, which is related and is used to electrically sense a constant magnetic field.

You're also correct that the resulting electric field is NOT due to Faraday's law. Indeed, Faraday's law only equates the curl $\nabla \times E$ with $\frac{\partial B}{\partial t}$. The electric field you see in your example (in the steady state) is due to the separation of charge, and is hence attributable to Gauss's law.

If you're wondering about the force that segregates the charges in the first place, that's the Lorentz force as you mention and is due to magnetic field rather than an electric field.

Answer 5

"My understanding of motional emf is that one of the ways it is created is by moving a conductor (moving such that its orientation doesn't change) in a uniform magnetic field"

If relationship that allows generation of an E-field from an uniform magnetic field is:

curl E = -∂B/∂t

Also it's difficult to generate an uniform magnetic field, can be done using Helmoltz Coils.

You mention "Of course, if you are moving a conductor in a field such that flux is changing (like changing the orientation of conductor), EMF is induced"

Yes you are right and quite often its not obvious why the magnetic field is changing. Consider this example (it took me a while to understand it)

Example how can a permanent magnet deflect an electron beam? It seems like there is no ∂B/∂t ? To understand lets again use:

curl E = -∂B/∂t

The magnetic field is changing (for the charged electrons) because the electrons are moving in what is a non uniform magnetic field so we do have ∂B/∂t and yes the electron beam will be deflected (so you have curl E) in fact its even possible to get the electron beam to spiral.