Helical Motion

Sections 10.2 and 10.3 assumed the charged particle's entire velocity lies in a plane PERPENDICULAR to the magnetic field $\vec{B}$, producing pure circular motion. What happens if the particle's velocity ALSO has a component $\vec{v}_{\parallel}$ directed ALONG $\vec{B}$ itself -- i.e. its motion is not confined to a single plane perpendicular to the field? The key is to treat the two velocity components entirely separately, since the magnetic force acts independently and linearly on each.

For the parallel component $\vec{v}_{\parallel}$, the magnetic force it alone would produce is

since $\vec{v}_{\parallel}$ is, by definition, exactly parallel to $\vec{B}$, and the angle between two parallel vectors is $0^\circ$ (for which $\sin0^\circ=0$). So the parallel component of the velocity experiences NO magnetic force at all -- it is completely unaffected by the field, and the particle simply drifts along the direction of $\vec{B}$ at whatever constant speed $v_{\parallel}$ it started with.

Meanwhile, the PERPENDICULAR component of the velocity, $\vec{v}_{\perp}$, behaves exactly as analysed in Section 10.3: it produces genuine circular motion in the plane perpendicular to $\vec{B}$, of radius $R=mv_{\perp}/(qB)$, exactly as before (using the perpendicular speed $v_{\perp}$ in place of the full speed $v$). …