ElectroMagnetics Lenz's law vs F=qv X B (2024)

Hello and thank you in advanced for looking at this. I have trying to come to terms with two seemingly contradictory ideas in electro-magnetics. Lenze's law and Eddie currents seem to contradict the fundamental equation F = qv X B where F is the force exerted on a particle with charge q and velocity v crossed with Magnetic field B and F, v, B are all vectors. I would like to illustrate this:

Suppose you have a copper plate, attached to the top and the bottom are the positive and negative terminals of a power supply respectively so current is flowing down (electrons going up).

Then you have two electro-magnets, one on each side with opposite polarities such that the flux lines flow through the plate.

When the magnets energizes the "F = qv X B" would imply that the electrons get accelerated toward one side of the plate and end up with a diagonal velocity (down from the electric field and sidways from the magnetic field).

But lenz's law, which as far as i can tell is solely responsible for eddy currents says that when those electric magnetic energizes the electrons will have to go in a circular motion to create a magnetic field which opposes the original field.

Is it the case that these are both true simultaneously and there will be swirling electrons and those swirls will move down in a diagonal direction? If not is there already an equation that has been driven that will tell me the total force on an electron taking both of these laws into account?

The discrepancy comes from the fact that "F = qv X B" is 0 when the velocity is zero, but lenzes law has a force on stationary electrons subject to a changing magnetic field to balance it out. Thank you in advance for any guidance I have thoroughly stumped myself :)

The force equation you want is: ##\vec F = q(\vec E + \vec v\times\vec B)## , which assumes that the fields do not change with time. Eddie currents and Lenz's law deal with the situation that the fields change with time.

Time-varying E and B fields create circulation in the force equation. (See: Maxwell's equations.)
This sort of complication means that you'll soon-enough start dealing with the behavior of charges in terms of energy rather than force.