B Force Exerted on a Conductor by a Homogeneous Magnetic Field

[Heisenberg7]

In my book it's said that a conductor in a homogeneous magnetic field moves because there is a stronger magnetic field on one side and a weaker magnetic field on the other. Now that seems wrong to me. I mean, if we were to look at the Lorentz force that the magnetic field exerts on the conductor, it should point in the same direction anyway (right hand rule; both sides). The way they say it, it's like "because of the difference in magnetic field/induction we get a force". So, is it because of the Lorentz force or the difference in magnetic field?

Thanks in advance.

 

[anuttarasammyak]

I think the textbook shows an interpretation of Lorentz force by magnetic field lines, i.e. vortex made by current around the wire should pushed away from dense to less dense side.