Why Does a Charged Particle's Kinetic Energy Stay Constant in a Magnetic Field?

A charged particle's kinetic energy stays constant because the magnetic force is always perpendicular to the particle's velocity. It changes the direction of the velocity, but not its magnitude, so the particle's speed and kinetic energy remain constant.

This is part of D.3 Motion in electromagnetic fields and applies at both SL and HL.

The Reasoning

The magnetic force on a particle of charge $q$, moving with velocity $\mathbf{v}$ through a magnetic field $\mathbf{B}$, has magnitude

$$F = qvB\sin\theta,$$

where $\theta$ is the angle between the velocity and the magnetic field. Its direction is perpendicular to both $\mathbf{v}$ and $\mathbf{B}$.

The rate at which a force transfers energy is its power:

$$P = \mathbf{F}\cdot\mathbf{v} = Fv\cos 90^\circ = 0.$$

Therefore, the magnetic force does no work:

$$W = \Delta E_{\text{k}} = 0.$$

For perpendicular entry, the magnetic force acts as the centripetal force that continually turns the particle toward the centre of its circular path. The acceleration is therefore centripetal rather than tangential: it changes the direction of motion without changing speed. For angled entry, the velocity component parallel to the field is unchanged, while the perpendicular component produces circular motion, giving a helix.

Since $E_{\text{k}} = \frac{1}{2}mv^2$, constant kinetic energy means constant speed. However, velocity is a vector, so it can still change when its direction changes.

A common misconception is that any force must change kinetic energy. This is false: only a force component parallel to the displacement can do work. An electric field may change the particle's kinetic energy, but a magnetic force alone cannot.

Exam Technique

For an Explain question, give the complete causal chain: magnetic force is perpendicular to velocity, so no work is done; therefore kinetic energy and speed remain constant, although velocity changes direction.