# Stumped on understanding a Feynman lecture about force from wire on magnet

I think your analysis is all good, except for your statement that $$\bf{B}$$ is in the $$y$$-direction.

This is true only for points that are directly beneath the wire.

Most points on your coil are not directly beneath the wire, they are off to the side a bit. At such points, the magnetic field from the wire has a vertical component, which gives a net force in the $$y$$ direction when crossed with the current direction at those points.

(Of course, there is also a y $$component$$ of B at every point on the coil, but the vertical force this causes is exactly cancelled by an opposite force acting on the symmetric point.)

Sounds like a strange request, he ? The answer you should accept is Paul G's since he posted it before mine.

I don't want to steal it, just make it maybe a bit clearer. I did it in comments, but it will be more visible here.

As Paul G wrote, because the field lines are circles and thus though the $$y$$ component of the $$B$$ field below the wire is the largest, there is a small $$z$$ component in the B field produced the wire at the position of the coils (which have finite radius), and the sign of this component is opposite on opposite sides of $$y=0$$ plane where the wire is. The current in the coils have opposite $$x$$ orientations on these opposite sides, so the relative sign with the $$z$$ component is the same so their cross products (which are in the $$y$$ direction !) add up to a net force in that direction.

Contrariwise, the $$y$$ component of B is the same on each side, and thus the (separately much larger !) cross products with the $$x$$ component of the current on each side of the plane exactly cancel each other by symmetry, so there is no net $$z$$ component to the force.

Again the net $$x$$ component of the force is zero. The $$z$$ component of the $$B$$ field do have cross products with the $$y$$ component of the current in the coils, but if you look at them carefully, you also find that they cancel by symmetry.