Helical Motion of Charged Particles

Suppose a uniform magnetic field $\vec{B}$ points along the $z$-axis, so $\vec{B} = B\hat{z}$. A particle of charge $q$ and mass $m$ enters with velocity $\vec{v}$ at an angle $\theta$ to $\vec{B}$. We decompose this into two independent components: the parallel component $v_{\parallel} = v\cos\theta$ (along $\hat{z}$) and the perpendicular component $v_{\perp} = v\sin\theta$ (in the $xy$-plane). These two components evolve completely independently because the magnetic force $\vec{F} = q\vec{v}\times\vec{B}$ acts only on the component of velocity perpendicular to $\vec{B}$.

The perpendicular component $v_{\perp}$ experiences a centripetal magnetic force. Setting the magnetic force equal to the centripetal force requirement: $|q|v_{\perp}B = \frac{mv_{\perp}^2}{r}$. Solving for the radius of the circular orbit (called the Larmor radius or cyclotron radius): $$r = \frac{mv_{\perp}}{|q|B}$$. The angular frequency of this circular motion is $\omega_c = \frac{|q|B}{m}$, known as the cyclotron frequency. Notice that $\omega_c$ does not depend on the speed of the particle — only on its charge-to-mass ratio and the field strength.

Because $\vec{B}$ exerts no force parallel to itself, the parallel component $v_{\parallel}$ remains constant throughout the motion — there is no acceleration in the $z$-direction. The particle therefore drifts steadily along the field line while simultaneously rotating. This combination of uniform circular motion in the $xy$-plane and uniform translation along $z$ is precisely what defines a helix.

The pitch $p$ of the helix is the distance the particle advances along the field direction during one complete revolution. The period of one revolution is $T = \frac{2\pi m}{|q|B}$ (again independent of speed). During this time the particle moves a distance $p = v_{\parallel}T$ along $\hat{z}$: $$p = \frac{2\pi m v_{\parallel}}{|q|B} = \frac{2\pi m v\cos\theta}{|q|B}$$. A large pitch means the helix is stretched out; a small pitch means it is tightly wound. When $\theta = 0°$, the pitch equals zero and the particle moves in a straight line. When $\theta = 90°$, the pitch is zero and the motion is purely circular.

Taking the particle to start at the origin with $v_{\perp}$ initially along $\hat{x}$, its position at time $t$ is: $x(t) = r\sin(\omega_c t)$, $y(t) = r(1 - \cos(\omega_c t))$, $z(t) = v_{\parallel}t$. These three equations fully specify the helical trajectory. The projection onto the $xy$-plane is a circle of radius $r$; the projection onto the $z$-axis shows steady linear motion. Together they paint the corkscrew path characteristic of helical motion in a magnetic field.

• r — Larmor (cyclotron) radius (m): Radius of the circular motion in the plane perpendicular to B
• m — Mass of particle (kg): Rest mass of the charged particle
• v_{\perp} — Perpendicular speed (m s⁻¹): Component of velocity perpendicular to the magnetic field
• |q| — Magnitude of charge (C): Absolute value of the particle's electric charge
• B — Magnetic field strength (T): Magnitude of the uniform magnetic field
• T — Period (s): Time for one complete revolution around the field line
• p — Pitch (m): Distance advanced along the field per revolution
• v_{\parallel} — Parallel speed (m s⁻¹): Component of velocity parallel to the magnetic field

Problem. A proton (mass $m_p = 1.67 \times 10^{-27}$ kg, charge $q = 1.60 \times 10^{-19}$ C) enters a uniform magnetic field $B = 0.50$ T with speed $v = 2.0 \times 10^6$ m s⁻¹ at an angle $\theta = 30°$ to the field direction. Find: (a) the Larmor radius, (b) the cyclotron period, and (c) the pitch of the helix.

Answer. Larmor radius ≈ 2.1 cm; Period ≈ 1.3 × 10⁻⁷ s; Pitch ≈ 23 cm

• Circular Motion of Charged Particles — Helical motion reduces to pure circular motion when the velocity is entirely perpendicular to B; this is the limiting case and must be understood first.
• The Lorentz Force — The magnetic component of the Lorentz force, F = qv×B, is the sole cause of both the circular and helical trajectories studied in this topic.
• Cyclotron Frequency and the Cyclotron — The cyclotron exploits the fact that the period of circular motion is independent of particle speed — the same key result derived in helical motion analysis.
• Magnetic Mirror Effect — In non-uniform fields, helically moving particles can be reflected by regions of stronger field — the magnetic mirror — which directly depends on the helical pitch and radius.
• Uniform Circular Motion — The centripetal acceleration framework used to derive the Larmor radius is identical to the analysis of uniform circular motion studied in mechanics.