A charged particle entering a magnetic field at an angle -- is work done?

[big_bounce]

Hello PF's members

I know a magnetic field doesn't work on charged particles if magnetic field being perpendicular to the velocity of the particles. also i know magnetic field doesn't work if a charged particle enters with right angle into the field.

But suppose that a charged particle like electron is entering with angel 35 degree at a uniform magnetic field.
Does magnetic field work on charged particle after entering and electron gains energy? what about when magnetic field is non-uniform?

 

[vanhees71]

A purely magnetic field never does work on the charged particle since the force is always perpendicular to the velocity,
$$\vec{F}_{\text{mag}}=\frac{q}{c} \vec{v} \times \vec{B} \; \Rightarrow \; P=\vec{v} \cdot \vec{F}=0.$$

 

[big_bounce]

A purely magnetic field never does work on the charged particle since the force is always perpendicular to the velocity,
$$\vec{F}_{\text{mag}}=\frac{q}{c} \vec{v} \times \vec{B} \; \Rightarrow \; P=\vec{v} \cdot \vec{F}=0.$$

EVEN If the electron enters the field at an angle to the field direction, magnetic force remains perpendicular to the velocity?

 

[jtbell]

Yes, the force is always perpendicular to both the field and the velocity.

 

[big_bounce]

therefore for changing motion of object in direction we always don't have to expend (or gain) energy?

 

[Tallus Bryne]

therefore for changing motion of object in direction we always don't have to expend (or gain) energy?

Think of a simple case of centripetal motion at some constant potential energy state (perhaps a "car" going around a circular, horizontal track at constant speed). The centripetal force is responsible for the change in motion, but that doesn't mean the energy changes (constant speed means constant kinetic energy).