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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.

Thanks in advance for the help.