# How does a force on electrons produce a force on a metal plate

In a paper from Cadwell Magnetic damping: Analysis of an eddy current brake using an airtrack about eddy current brakes the author explains the effect qualitatively as follows:

When a nonferromagnetic conductor passes between the poles of a magnet, an electric field is induced, and circulating currents called eddy currents are generated. As a result, a magnetic damping force is induced on the eddy currents which opposes the motion of the conductor.

As an aluminium plate enters an external magnetic field [...] a Lorentz force [...] is exerted on the conduction electrons in the aluminium plate[...] The velocity of the plate $v$ and the magnetic field $B$ are orthogonal to one another. An induced current moves along a closed path [...].

A horizontal magnetic force is exerted on the portion of the eddy current that is within the magnetic field. This force is transmitted to the aluminium plate, and is the retarding force associated with the magnetic braking.

(my emphasis)

The part about the moving electrons and the Lorentz-force is pretty clear. But how and why is the force transmitted to the aluminium plate? I mean we are speaking of conduction electrons, so I just would expect that the Lorentz force alters the path of the electrons, but why does the Lorentz force on those conduction electrons affect the aluminium plate as a whole?

And why is the Lorentz-Force which is due to the motion of the aluminium plate with respect to the magnetic field not transferred to the plate as a whole (resulting in a force orthogonal to the direction of motion on the whole plate)?

Edit

Here is an illustration of the set up from the paper:

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When a car starts accelerating, with the windows closed, we do not feel a backwards breeze: The air accelerates more-or-less in sync with the car. Why? After all, there are no rubber bands connecting the air molecules to the car frame. How do the air molecules know to start moving when the frame starts moving?

Scattering means that when, say, the car seat moves through the initially-stationary air, it pushes on the air, causing the air to move ever closer to the same rate that the car frame is moving. In a similar way, every "obstacle" in the car pushes on the air surrounding it.

Confinement is the effect of the rear windshield. When air starts to "pile up" at the rear windshield, the pressure there gets higher. If the pressure is higher at the back of the car than the center, then that creates a force that pushes forward on (eventually) all the air in the car. (There is an analogous effect at the other end, where the front windshield leaves behind a lower-pressure zone behind it, causing the air to move forward faster.)

The net effect is that at normal accelerations, the air accelerates in sync with the car.

That brings us to solids. A force on the atoms in a solid gets (almost) immediately transmitted to the electrons in it, and a force on the electrons gets (almost) immediately transmitted to the atoms. Why? Same as above: Scattering and confinement.

Scattering means that electrons are constantly moving around, and frequently collide with atoms. If the electrons and atoms are moving (on average) with respect to each other, they will "drag" on each other, which has the effect of tending to bring the velocities into sync. (This "drag", of course, is responsible for electrical resistance.)

Confinement means that electrons are stuck in the metal. You may notice that electrons don't go frequently flying out of pieces of metal that are just sitting around. The reason that they don't do that is because that there's a significant force, quantified by the workfunction, keeping them in the metal. Basically, the electrons are attracted to the nuclei in the metal. So if the electrons and atoms are moving with respect to each other, the electrons will pile up at one end and empty out at the other end, which in turn creates a force counteracting this pile-up. Again, the effect is to bring the velocities of the electrons and atoms into sync.

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+1 for being much much much much much much much much much much better than my answer – Jim Apr 30 '13 at 13:00

Here's an initial brief answer. Perhaps a better answer will come along later.

You understand that from the perspective of the aluminium plate, the magnetic field due to the magnets is changing over time. And you understand that this changing magnetic field generates an electric field within the plate. This electric field causes free electrons within the plate to loop around the field lines. This motion of the electrons generates a magnetic field that directly opposes the direction of change of the external magnetic field. Because the force on the electrons is not strong enough to tear them from the aluminium plate, and because (in the big picture) the induced mag field is coming from the plate, the net effect is that the aluminium plate moving through the external field creates a new field opposing the motion.

To summarize this in a way that directly answers the question, the force is transmitted to the plate because the electrons are part of the plate. By imposing a force on them, one imposes a force on the plate.

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Thanks, but that's a bit too hand-waving to me. – student Apr 29 '13 at 16:42
I don't see for example how the eddy current in the illustration I added in my last edit "knows" that he is "captured" in the plate somehow. So I need really more details of how the moving conduction electrons interact with the plate. – student Apr 29 '13 at 16:51
the moving conduction electrons, or "free" electrons, are only free in the sense that they can move from atom to atom at will within the metal plate. However, they are still bounded to the structure overall by the positive charges in all of the nuclei. So while the eddy current can freely move within the plate, all of those electrons are still bound through coulombic forces to the plate. – Jim Apr 29 '13 at 17:09
Can this understood really in a classical sense? I think this touches the metallic bonding theory which seems to be a quantum effect, see for example: th.fhi-berlin.mpg.de/th/publications/… Sec. 3.2. – student Apr 29 '13 at 17:37
Relating to your last comment: Why is then the Lorentz-Force which is due to the motion of the aluminium plate with respect to the magnetic field not transferred to the plate as a whole (resulting in a force orthogonal to the direction of motion on the whole plate)? – student Apr 29 '13 at 17:38