I understand that, with an electron moving perpendicular to a magnetic field, the Lorentz force acting on it will be perpendicular to both the velocity vector and the magnetic field. Hence the force cannot change the magnitude of the velocity, only its direction, and the electron moves in a circular path.

But then later in my book it moves on to discuss the force on a current-carrying piece of wire due to a magnetic field. I follow the derivation of this force, and it ends up with \(\vec{F}=I\vec{L}\times\vec{B}\). I thought I understood this okay, but then after more thought something is puzzling me. Imagine I have some space with a B field constant over it, and into this I put a straight piece of wire of mass m, carrying constant current I perpendicular to the B field. The above force will act upon the wire and accelerate it, as a whole, in a direction perpendicular to the current and the B field. So is it not the case here that the B field is doing work on the wire?

Can someone please help me to understand this, as the conclusions of the above paragraphs seem conflicting.

Many thanks!

Jon.