Show that a charged particle moving at right angles to a uniform magnetic field moves in a circle. Derive expressions for the radius and the frequency, and explain why the frequency does not depend on the speed.

1. 1 · State the force. F = q(v × B): size qvB when v ⊥ B, always perpendicular to v.
2. 2 · Draw the path. [Crosses ⊗ for B into the page; a circle with v along the tangent and F pointing to the centre; label q and r]
3. 3 · Equate forces. The magnetic force is the centripetal force: mv²/r = qvB, so r = mv/qB.
4. 4 · Find the period and frequency. T = 2πr/v = 2πm/qB; ν = qB/2πm.
5. 5 · Explain. r ∝ v, so a faster particle covers a bigger circle in the same time: ν has no v in it.
6. 6 · Close in one line. So, the magnetic force only turns the particle; it does no work and the speed stays constant.

Scoring tip: Write 'force perpendicular to velocity, so no work is done'. This one line earns the mark for explaining why the path is a circle.