# Is electromagnetic induction a result of the Hall effect?

I know that an EMF is induced when there's a change in magnetic flux through a conducting surface. Is this due to the Hall effect?

By this I mean, do the delocalised electrons in the conductor experience force due to the external changing magnetic flux, and thus accumulate on one side of the conductor, causing a difference in potential ---> an EMF (and subsequently a flow of current if the conductor is part of a closed loop).

If this is not the case, then how exactly is EMF induced by the changing magnetic flux?

The two cases are unrelated. The Hall effect is a response of a conductive medium to the Lorentz force. The charge carriers redistribute in the conductor in such a manner that that an electric field is created that cancels the magnetic force. The total force on the carriers is then zero.

On the other hand, a time dependent magnetic field is always accompanied, following the Maxell-Faraday equation, by an electric field such that $\vec \nabla \times \vec E = -\frac{d\vec B}{dt}$. This is independent of whether there is a conductive medium present or not. If there is, the medium will respond by a current such that $\vec J = \sigma \vec E$. If there is no closed loop, a time-dependent charge difference will result such that $\vec E$ is cancelled.

When a magnetic field is applied to a (already) current carrying conductor or semiconductor, an electric field is developed across it, in a direction perpendicular to both the current and the magnetic field. And that's your Hall effect. The difference here with Electromagnetic induction is that, the current isn't a byproduct of the changing magnetic flux. In fact the magnetic flux isn't changing at all. It's a constant current and a constant magnetic field which shall give you electric field within the material. This happens as the the following relation is true, $$F=q(E+\vec{v}\times\vec{B})$$ Therefore the charge particle flowing through the material would naturally get forced towards a direction perpendicular to both the motion and the magnetic field. This force however would quickly die out as a steady state is reached, that gives us, $$E=-(\vec{v}\times\vec{B})$$ This electric field is basically due to the uneven distribution of the charge particle throughout the body, due to the force working. Again the steady state is reached because the Hall field, exerts a force on the charge particles in the opposite direction. In case of Electromagnetic induction, the electric field is given by, $$\mathcal{E}=-\frac{d{\Phi}_B}{dt}$$ Where $\mathcal{E}$ is the EMF and ${\Phi}_B$ is the magnetic flux. One can get that from Maxwell's equations (specifically Maxwell-Faraday) as well. However induction has its application in Hall effect sensors. More about which can be found here.

They are different in the following ways:

1. In Hall effect, the magnetic field can be a constant over time, and its magnitude is related to the eventual electric field strength; but in EMF, it must change, and has a non-zero time derivative, and only its rate of changing is related the induced electric field strength.

2. In Hall effect, the electric filed due to charge accumulation has sources -- you can find the beginning and the end of the field lines, and $E$ field is static after equilibrium; in EMF, the electric filed is sourceless hence is not static by definition, and filed lines form closed loops.

3. In Hall effect the force on charges is from $H$ filed, and is a constant; while in EMF the force on charges is not directly from $H$ filed, but from the electric filed induced by the change of $H$ field, and the force is changing (at least direction).

4. In Hall effect, charge motion first and $E$ field follows: charge moves due to Lorentz force --- no $E$ field appears, then if the material has boundaries to accumulate charges, $E$ field appears; in EMF the procedure is reversed: $E$ appears first and charge motion follows: firstly $E$ field is induced by the change of $H$ --- no charge required at all, then if one puts charge in, it would go into a circular-like motion.

More formally, EMF appears since the $H$ or $B$ field is changing, in one equation: \begin{equation} \vec{\mathcal{E}} = \frac{d\vec{\Phi}}{dt} \propto \frac{d|B|}{dt} \end{equation} No charges yet. And if charges exits, its acceleration: \begin{equation} \vec{a}_e \propto \vec{\mathcal{E}} \end{equation} While in Hall effect, charges experience Lorentz force, and its acceleration: \begin{equation} \vec{a}_e \propto q_e\vec{v}\times\vec{B} \end{equation} No $E$ field yet. And if boundary exist for charge accumulation, after equilibrium: \begin{equation} \vec{E} \propto \vec{v}\times\vec{B} \end{equation} Hence EMF is a purely dynamical effect, while Hall effect is a purely static one --- by static and dynamic I mean time-dependence of $H$ field.

(I'm assuming a high-school level answer is proper, if you want more modern perspective, feel free to comment.)

It isn't. You can see that from the fact that the Hall coefficient depends on the magnitude of the field components perpendicular to the field, while electromagnetic induction only depends on the rate of change. Why should it be so, if the Hall effect caused the induction? Also, induction can happen anywhere, even in vacuum, whereas the Hall effect happens in conductors (well, it also happens in other kinds of media, but the point remains that it is related to the material). If the former were caused by the latter, it shouldn't happen in situations in which the effect isn't present, and it should become more and more relevant whenever the Hall effect becomes important.

Another point, somewhat more subtle, is that, while quantum mechanics is needed for any satisfactory explanation of the Hall effect, it is, as far as I know, perfectly consistent with the semiclassical approximation, in which the electromagnetic fields are classical. Thus, it makes sense within the framework of Maxwell's theory, in which electromagnetic induction is very fundamental, for it is a very direct consequence of Maxwell equations. By proving it, one is basically proving one or two of those equations. But then, how did you get the fields in the first place? You need a magnetic field in the region from the start. How do you produce it? When you measure it, what calculations are you making? What about the electric field? The point is, you are going to assume Maxwell's theory at some point (or some more fundamental theory, but in that case it has to explain the Maxwell equations). So the Hall effect couldn't offer a satisfactory explanation for induction, for any such argument must be circular.