Helical Motion — Pitch

When v has a component parallel to B, that component is unaffected by B. The particle traces a helix: radius r = mv⊥/qB from the perpendicular component, pitch p from the parallel component.

Setup

A charged particle enters a uniform field $\vec{B}$ (along $z$-axis) with velocity $\vec{v}$ at angle $\alpha$ to $\vec{B}$.

Resolve:

• $v_{\parallel} = v\cos\alpha$ — component along $\vec{B}$
• $v_{\perp} = v\sin\alpha$ — component perpendicular to $\vec{B}$

Motion in Each Component

Perpendicular component: $\vec{B}$ exerts force, causing circular motion in the $xy$-plane.

$$r = \frac{mv_{\perp}}{qB} = \frac{mv\sin\alpha}{qB}$$

Parallel component: $\vec{B}$ exerts no force on $v_{\parallel}$ (since $v_{\parallel} \parallel \vec{B}$, $\sin 0° = 0$). The particle moves at constant $v_{\parallel}$ along $z$.

Helical Path

The combination of circular motion in $xy$ and uniform motion in $z$ gives a helix.

Period of one circular revolution: $T = 2\pi m / qB$

Distance advanced along $z$ in one revolution — the pitch:

$$\boxed{p = v_{\parallel}\,T = \frac{2\pi m\,v_{\parallel}}{qB} = \frac{2\pi m v\cos\alpha}{qB}}$$

Summary