Cyclotron Motion

A key structural consequence of Section 10.2 is that the magnetic force on a moving charge is always perpendicular to its velocity, and always has a constant magnitude for a fixed speed and field strength. This is exactly the condition needed to produce UNIFORM CIRCULAR MOTION -- a net force of constant magnitude, always directed towards a fixed centre, is precisely what is needed to keep a particle moving in a circle at constant speed (a fact already familiar from the study of circular motion). Consider, then, a charged particle of charge $q$ moving with speed $v$ in a region where a uniform magnetic field $\vec{B}$ is directed perpendicular to the plane of the particle's motion (Fig. 10.5). The magnetic force on the particle, $qvB$ (since $\vec{v}\perp\vec{B}$ here, so $\sin\theta=1$), acts at every instant towards the centre of a circle of some radius $R$, and this force supplies EXACTLY the centripetal force needed to sustain that circular motion:

Rearranging this single equation gives two equally important results. First, solving for $R$:

Second, rewriting it in terms of the particle's momentum $p=mv$:

This last relation, $p=qBR$, is known as the cyclotron formula, and it is the mathematical foundation of the cyclotron accelerator discussed in the next subsection: it says that a charged particle of a given momentum, moving perpendicular to a given magnetic field, traces out a circular arc of a radius fixed entirely by that momentum, charge and field -- and, conversely, that measuring the radius of the circular path lets you determine the particle's momentum (or, if the momentum is known, its charge-to-mass ratio) directly. …