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

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## Main Question or Discussion Point

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?

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vanhees71
Gold Member
2019 Award
A purely magnetic field never does work on the charged particle since the force is always perpendicular to the velocity,
$$\vec{F}_{\text{mag}}=\frac{q}{c} \vec{v} \times \vec{B} \; \Rightarrow \; P=\vec{v} \cdot \vec{F}=0.$$

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

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

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).