Cyclotron Motion & the Cross-Field Cycloid

Equations

Uniform \(\mathbf{B} = B\hat{k}\): circular gyration at the cyclotron frequency with Larmor radius

\[\boxed{\,\omega_c = \frac{qB}{m}\,} \qquad \boxed{\,r_L = \frac{m v_\perp}{qB}\,}\]

Add \(\mathbf{E} = E\hat{\jmath}\) (crossed fields) and a particle starting at rest traces a cycloid:

\[x(t) = R(\omega t - \sin\omega t), \qquad y(t) = R(1 - \cos\omega t), \qquad R = \frac{mE}{qB^2}\]

— gyration superposed on the E×B drift \(v_E = E/B\) (the rolling-wheel curve: the wheel's center moves at \(v_E\)).

Physical meaning

The two fundamental scales of magnetized-particle motion: how fast it circles (\(\omega_c\)) and how big the circle is (\(r_L\)). Their smallness compared to field scales is what licenses the entire guiding-center approximation.

Variables

\(q, m\) — charge, mass · \(B\) — field strength · \(v_\perp\) — speed ⊥ \(\mathbf{B}\) · \(E\) — crossed electric field · \(R\) — cycloid radius.

Why it's the course benchmark

The cycloid is an exact, nontrivial solution of the Lorentz force: oscillatory (stresses stability), analytic (permits exact error measurement), and physically meaningful (it is the E×B drift). Every integrator in Chapter 2 is graded against it — see convergence and error and the worked example.

Numbers worth knowing

Electron in a 1 T field: \(\omega_c/2\pi \approx 28\) GHz (microwaves — the basis of electron-cyclotron heating). Proton in Earth's field (\(\sim 50\ \mu\)T): \(\omega_c/2\pi \approx 0.76\) Hz.

Lorentz force — parent
E×B drift — the cycloid's average velocity
Leapfrog update — reproduces gyration exactly in radius

Quiz

Q1 (computational). For \(q = m = E = B = 1\) (normalized), what are \(\omega\), \(R\), and the drift speed?

Answer

\(\omega = 1\), \(R = 1\), \(v_E = E/B = 1\) — the course's standard test configuration.

Q2 (conceptual). Ions and electrons gyrate in opposite senses. Why does the cycloid's drift direction nonetheless come out the same for both?