When an electron with charge $q$ travels with velocity $v$ perpendicular to a magnetic field generated between two permanent magnets with field strength $B$ and no electric field, it experiences a Lorentz force equal to $$F = qv \times B$$ The resulting change in momentum for the electron will be transferred through the magnetic field to the magnets. For example in a setup like this:
the electron would experience a change in momentum upwards and the magnets would experience an equal and opposite change in momentum downwards, due to conservation of momentum.
My question is, does this "reaction force" on the magnets also apply when you have a magnetic field inside a magnetic field, such that the superposition of the two fields results in no magnetic field at the electron's position. For example, lets say you have this following thought experiment with 4 magnets and a moving electron:
Where the red oval represents zero (or essentially zero) magnetic field. The magnetic field lines pointing right from the small magnets exactly cancels out with the magnetic field lines pointing left from the big magnets. If you just had the big magnets the electron would experience a force down (into the screen), and if you just had the small magnets the electron would experience a force up (out of the screen), but these cancel out, so there is no net force on the electron.
If you think there must still be a field between the two small magnets, either mentally increase the strength of the two larger magnets or mentally move the smaller magnets further apart. Here is a zoomed in visualization of the magnetic field lines to help visualize this. Sorry for the image quality:
This might seem counter intuitive, but it is possible because the strength of a magnetic field is proportional to the inverse cube of distance to the magnet. Imagine how a compass still points to earth's magnetic north, even if it is between 2 magnets spaced 100 meters apart.
So which of the following happens?
1) The big magnets experience a change in momentum up and the small magnets experience a change in momentum down.
2) None of the magnets experience a change in momentum. This would be the case if the experiment results in the magnets not moving.
- There is no violation of conservation of momentum here, in both 1 and 2, the net momentum change is 0.
- I expect there is a knowable concrete answer (1 or 2) because this could be tested in the real world with a relatively uncomplicated experiment.
- I am more interested in a concrete 1 or 2 and less interested in a why, but a general reasoning of why would be nice. I won't be able to follow a math explanation if it uses more than simple derivatives or integrals.
- I tried to look for duplicate questions that might already answer this, and there are a lot of related questions, but I couldn't conclude a concrete answer to this question from them. The closest I found was this question, which might hold the math to get the answer, but unfortunately I couldn't follow all the details.