What is the force on a moving charged particle in a magnetic field, and why is no work done by this force when it accelerates the particle?

A particle with electric charge q, mass m and velocity v in a constant magnetic field B experiences a force due to the field:

F = qv X B

The 'X' (cross-product) means that the force is always at 90° (perpendicular) to both the velocity and magnetic field.

If v and B point along the same line (parallel or antiparallel), the particle feels no force. Otherwise it will feel a force and accelerate (change in velocity). But acceleration doesn't necessarily mean change in speed.

Because the force is perpendicular to the velocity (direction of travel), the particle's velocity changes only in direction. The speed (magnitude of the velocity) of the particle does not change.

 

Work is defined as the force F applied to the particle times the distance s over which is it applied in the same direction.

W = Fs

(technically, W = ∫ F • ds but they took the fun out of A level physics, eh?)

Since the magnetic force is always perpendicular to the direction of travel, and hence only changes the particle's direction and not its speed, no work is done on the particle by this force:

W = 0.

N.B. The units Joules and Newton-metres are equivalent. [J] ≡ [Nm]

 

To understand further, try googling 'vector cross product', 'vector dot product'. The magnetic force is a cross product. The definition of work is a dot product. Can you guess why they are called that? :P