# Static Magnetic Field and Work

1. ### I done know

5
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.

2. ### Physics Monkey

1,358
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?

3. ### I done know

5
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.

4. ### Physics Monkey

1,358
Wait, how did you conclude that $$\frac{d\vec{v}}{dt} = 0$$? The acceleration certainly isn't zero, there is a force acting.

5. ### I done know

5
Is it that the force is perpendicular to the magnetic field and work must be parallel to the displacement?

6. ### Physics Monkey

1,358
You are so close! The force is perpendicular to the field, but that's not what matters. What else is the force perpendicular to?

7. ### I done know

5
Ah, so because the force is perpendicular to the velocity, the force is perpendicular to the displacement.

8. ### Physics Monkey

1,358
Indeed. In simple terms, the power $$\vec{F} \cdot \vec{v}$$ is identically zero. Hence no work is done.