Static Magnetic Field and Work

I done know

Why can't a static magnetic field (not changing in time) ever do work? How do I express this formulaically? My only guess is that work is zero for a closed path.

Physics Monkey

The force on a charged particle in a magnetic field is [tex] \vec{F} = q \vec{v} \times \vec{B} [/tex], right? Stare at that formula for a bit. Now ask yourself, how is the force related to the velocity? Then ask, how is work related to force?

I done know

I see, then dv/dt is 0 when the B field is static, so if a=0 then F=0 then W=0. Sound right?

Unfortunately this was on our last exam, and my answer was that W=qV and induced voltage is only a result of B flux changing in time.

Physics Monkey

Wait, how did you conclude that [tex] \frac{d\vec{v}}{dt} = 0 [/tex]? The acceleration certainly isn't zero, there is a force acting.

I done know

Is it that the force is perpendicular to the magnetic field and work must be parallel to the displacement?

Physics Monkey

You are so close! The force is perpendicular to the field, but that's not what matters. What else is the force perpendicular to?

I done know

Ah, so because the force is perpendicular to the velocity, the force is perpendicular to the displacement.

Physics Monkey

Indeed. In simple terms, the power [tex] \vec{F} \cdot \vec{v} [/tex] is identically zero. Hence no work is done.