Eddy Currents and Lenz's law

Question

I am a TA in an undergraduate physics class. Part of the lab on magnetism has the students moving an aluminum metal sheet through a magnetic field as shown in the image below. The plate passes between a "north pole" and a "south pole". One side of the sheet has slits cut into it, the other end does not. The students are supposed to feel a "resistance" when they try to pull the solid side through the field, but they will not feel that "resistance" when they use the cut side.

Of course we can understand this through the Eddy currents. The initial motion of the plate through the field will cause the electrons in the plate to obtain a vertical velocity component from the magnetic force (whether it is up or down depends on the direction of the field and which way you are pulling the plate). This vertical velocity component will then cause the magnetic force to push the electrons in the opposite direction you are pulling them. The cut side prevents these eddy currents from forming (or at least being as large) and so you do not feel the "resistance". This is explanation is fine for the purposes of the lab.

However I was wondering about thinking of this in terms of Lenz's law. It makes sense that we have a changing magnetic flux through the plate as we bring it into/out of the magnet, and so a current will be produced in the plate to fight this changing flux. I verified this by placing a bunch of these magnets next to each other and moving the plate through all of them. Sure enough, when the plate is fully in the magnets I no longer feel a resistance to my force. Only when the plate is moving in/out of the magnets do I feel the resistance.

Then I started thinking about the cut side with the single magnet. While I am bringing the plate in, the change in flux is exactly the same as in the solid side when a slit is not at the magnetic field boundary. So over all the change in flux will look like the solid side but with periods of time where the change in flux is 0.

My question: Is the step like nature in the overall change in flux over time enough to explain why we do not feel a resistance on the cut side? Or is Lenz's law in this case more suited for in which "direction" our "resistance" is in, and we need to bring in other machinery to determine how large this resistance will actually be?

Answer 1

"when the plate is fully in the magnets I no longer feel a resistance to my force"

This surprises me, unless the plate is fully immersed in a homogeneous field. If the field varies over the area of the conductor I'd expect resistance to be felt. This is because there will be an emf induced in the part of the plate in the strongest (perpendicular) field, and any circuit path through the plate will pass through regions of weaker emf, so there will be a net emf in the path. There will therefore be eddy currents and resistive forces.

I've avoided talking about changes in flux linkage; arguments based on this can be treacherous if the circuit isn't a single wire loop. I'm a little worried about your approach, as in "It makes sense that we have a changing magnetic flux through the plate as we bring it into/out of the magnet […]" This isn't the only circumstance when there will be an emf. You have to consider various possible loops that the current could take (imagine drawing them with a pencil on the metal sheet). Then consider whether the flux through any loop changes as the sheet moves between the magnet's poles. If it does change, there'll be a current in that loop. An edge of the sheet doesn't have to enter or leave the magnet's field in order for there to be current.

As for why no resistance felt for the cut side… Theere will still be induced emfs, just as for the uncut side. Remember that you don't need to have a conductor in order to have an emf; you just need a path through space, around which work would be done on a charge. The point is that the slots prevent the emfs driving currents round any decent-sized paths.