Solving the Paradox of Electromagnetic Induction

[Coffee_]

Hello everyone, I have the following problem understanding electromagnetic induction. THis is not a homework question but it's a thought experiment I seem to not succeed in doing.

Let's take the situation where you move a rectangle shaped plate with a speed perpendicular to a homogenous magnetic field. Now for 2 different reasonings I get different results:

Lorentz Force
Because the electrons in that plate, have a speed due to the moving of the plate, they are moving particles in a magnetic field, therefore there is a lorentz force on them. In class we expressed that as F=B*Q*v. If you keep moving the plate, all the electrons will move to one side of the plate (depending on your arrows) making that side negatively charged, and the opposite side positively charged. You have a difference in potentials, therefore a voltage.

Electromagnetic induction
If we look at that plate as having a flux through it, we can say that because it is a homogenous field, as long as the speed is constant the flux through the plate will be exactly the same for every moment. So when moving a certain distance through this field we can look at what the voltage is. Well Δflux = 0 because at every moment the flux is the same, so the induced voltage is as well zero.

Obviously this is not right, I assume I made a reasoning mistake in the electromagnetic induction part, where is it? Thanks a lot.

 

[Steely Dan]

There is no mistake in your reasoning; the mistake is in conflating two different types of electric fields. What you have described in the first part is commonly called the Hall effect, and describes an electric field transverse to the motion of the electrons. What you would have obtained as a result of the induction equation is an electric field parallel to the motion of the electrons, had there been a changing flux. In other words, the induced potential difference because of changing magnetic flux is not the only way to achieve a potential difference across the conductor; in this case, the other potential difference is simply perpendicular.

 

[Coffee_]

There is no mistake in your reasoning; the mistake is in conflating two different types of electric fields. What you have described in the first part is commonly called the Hall effect, and describes an electric field transverse to the motion of the electrons. What you would have obtained as a result of the induction equation is an electric field parallel to the motion of the electrons, had there been a changing flux. In other words, the induced potential difference because of changing magnetic flux is not the only way to achieve a potential difference across the conductor; in this case, the other potential difference is simply perpendicular.

Hmm, I see what you are saying, it is weird then that in my textbook the electromagnetic induction is explained using the induction caused by the lorentzforce. They are set equal to each other, I even thought the teacher said that the electromagnetic induction actually is the induction due to the lorentzforce, or my memory must really be playing a trick on me.