I'm trying to understand how electric generators work on a very basic level. I understand the heart of this is the application of the Lorentz force on a conductor moving at a velocity ##v## in the presence of a magnetic field ##B##. I also understand how it can be shown that the emf is equal to ##-Blv## where ##l## is the length of the conductor.

Now, this emf corresponds to the separation of charges (positive charges pile up at the top and negative charges pile up at the bottom) due to the Lorentz force, which is then balanced by the Coulombic attraction between the positive and negative charges. I also understand that the negative sign comes from Lenz's law, which says that this voltage that is developed is intended to move current in the conductor to produce an induced magnetic field so that it opposes the change in the magnetic field through the conductor.

I would expect the top of the conductor to be more positive relative to the bottom. But if ##V = -Blv##, then the voltage at the top is more negative relative to the bottom? That part confuses me. What confuses me further is that this negative sign is commonly discarded. Consider for example, this at 7 minutes.

Why is this allowed? Isn't the negative sign crucial to labeling how charges move through the conductor and ultimately through a generator? I know that a generator involves multiple linked segments of conductors that push the charges around a loop with two sides being responsible for the total voltage being developed at the terminals of the generator. So, I would think preserving the minus sign in calculating the total voltage generated at the terminals of a generator would be important.

I want to show a diagram of my own that really illustrates the heart of what I'm confused about.

Basically, the Lorentz force should make positive charges go up top, but ##V_{\ell} = -Bv\ell## implies that the top is more negative than the bottom. How is this possible? Where does the contradiction come from?

No , it doesn't . When you move along the direction of electric field , does potential drop increase or decrease ?
Hint : - ∫ E.dl = ΔV . Check your integration in your second post .

As you move along the direction of the electric field, the potential drops because the field is doing work on you. I am not sure what is wrong with the integration.

Why is it ##V_0 - V_l## and not ##V_i - V_0##? I thought the definition of voltage was
[tex]V_{\text{final}} - V_{\text{initial}} = -\int_{\text{initial}}^{\text{final}} E \cdot dl[/tex]

That makes perfect sense. That is where the extra negative sign comes from. So, that means ##V_{\ell} = B\ell v## in the end. I wonder if those teachers / professors in those YouTube videos / lectures were all assuming something about the B field and just forgetting to include it? It feels like misinformation to say ##V_{\ell} = -B\ell v## then.