Why Can't a Static Magnetic Field Do 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 \vec{F} = q \vec{v} \times \vec{B}, 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 \frac{d\vec{v}}{dt} = 0? 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 \vec{F} \cdot \vec{v} is identically zero. Hence no work is done.