I am a tutor and during a session today, a thought occurred to me. I've long been told (and been telling) that magnetic fields do no work, but it's pretty easy to imagine a situation in which a magnetic field DOES do work. If two wires have parallel currents traveling through them, with one wire situated above the other, the magnetic force would lift up the lower wire.
This would be basically the definition of work.
Why doesn't this negate the statement that magnetic forces do no work?
Magnetic fields do no work on unconstrained charged particles. This comes from the Lorentz force law: $$F = q\vec{v} \times \vec{B}$$ \begin{align} dW &= \vec{F}\cdot{}d\vec{x} \\ &= (q\vec{v} \times \vec{B})\cdot{}d\vec{x} \\ &= (q\vec{v} \times \vec{B})\cdot{}\vec{v}\,dt \\ &= 0 \end{align} (The last step is due to $(\vec{x} \times \vec{y})\cdot\vec{x} = 0$ for any vectors $\vec{x}$ and $\vec{y}$.)
It is certainly possible for magnetic fields to do work on other things:
It is supprising how many people confuse work done with energy. A static magnetic field contains no energy -and it is why magnet only based examples of perpectual motion machines do not work . Magnetism is a force only, and work is done when a mass is moved a distance by that force. Clearly the magnetic force is known to move objects within its field of influence,and is capable of doing work as you describe. Some might be convinced that because work is done then energy must be available, however the only energy present is that which was first applied (eg the power x time for which the parrallel conductors carrying current) or the energy required to place an object into the field of influence in the first place. The energy is either returned ( ideally) or converted into mechanical movement ( as per an electric motor). Static magnetic fields are not a source of energy.
