Cyclotron Motion and Magnetic Confinement

The Lorentz force law handed us a strange instrument: a force that grips every moving charge yet refuses, on geometric principle, to do any work. What motion does such a force actually produce? Release a charge into a uniform magnetic field and watch. The answer is a machine part of astonishing regularity — a circle whose period does not depend on how fast the particle moves. A slow proton crawls around a small circle, a fast one races around a big circle, and both come home in exactly the same time, $T = 2\pi m/qB$ . That coincidence, scribbled on a library note by Ernest Lawrence in 1929, became the cyclotron, and with it the entire technology of particle accelerators. This article solves the motion completely. A uniform field splits the velocity into a component along the field, which coasts untouched, and a component across it, which wheels around at the cyclotron frequency $\omega_c = qB/m$ — the generic trajectory is a helix threading a field line like a bead on a corkscrew. Add a crossed electric field and the circle's center itself glides sideways at $\vec E\times\vec B/B^2$ , indifferent to the particle's charge, mass, or sign — the drift that organizes all of plasma physics. Then we cash the title's second promise: confinement. A uniform field ties particles to field lines but lets them stream freely along them; real bottles — fusion mirrors, tokamaks, the Van Allen belts wrapped around Earth — plug the ends with stronger field, exploiting a quantity, the magnetic moment $\mu = mv_\perp^2/2B$ , that gyration holds nearly constant while the field slowly changes. Why hospital cyclotrons top out near tens of MeV, why the aurora hangs in rings around the poles, and why fusion ions circle on millimetre leashes: all of it is this one solved trajectory, dressed three ways.

Key takeaways

With only a uniform $\vec B$ acting, speed is constant (no-work theorem, recalled from Lorentz Force Law); the velocity splits into $v_\parallel$ (untouched) and $v_\perp$ (rotating).
The perpendicular motion is a circle of radius $r_c = mv_\perp/qB$ traversed at the cyclotron frequency $\omega_c = qB/m$ — independent of speed: fast particles ride proportionally bigger circles and keep the same period $T = 2\pi m/qB$ .
The general orbit is a helix along the field line, with pitch $2\pi v_\parallel/\omega_c$ ; injection exactly along $\vec B$ feels no force at all.
Isochronism powers the classical cyclotron: one fixed RF frequency kicks the spiralling particle every half-turn, to $K_{\max} = q^2B^2R^2/2m$ — until relativity breaks the clock ( $\omega = qB/\gamma m$ ) and synchrotrons take over.
Crossed fields make the guiding center drift at $\vec v_d = \vec E\times\vec B/B^2$ — independent of $q$ , $m$ , and sign, so entire plasmas drift together.
Confinement along the field needs field gradients: the adiabatic invariant $\mu = mv_\perp^2/2B$ turns strong-field regions into magnetic mirrors, with a loss cone $\sin^2\theta_{\rm loss} = B_0/B_{\rm max}$ .
KEY IDEA: in a uniform field a charge gyrates at $\omega_c = qB/m$ on a circle of radius $mv_\perp/qB$ while coasting along the field — the speed cancels from the clock, and that isochronism built the accelerator age.

The core insight

In a uniform $\vec B$ the magnetic force does no work; velocity splits into free drift along $\vec B$ plus circular gyration at frequency $\omega_c = qB/m$ with radius $r_c = mv_\perp/qB$ — a helix whose orbital period is independent of speed.

Where this lives today

Count the cyclotrons before counting the colliders: some 1,500 medical cyclotrons produce PET isotopes and proton-therapy beams worldwide, all running equation (6) at the tens-of-MeV scale of worked example 2. The research frontier kept the name but dropped the fixed clock — from PSI's 590 MeV ring cyclotron to the LHC, every modern machine is a variation on "hold the orbit, sweep the parameters," engineering around the $\gamma$ in $\omega = qB/\gamma m$ . Penning-trap mass spectrometry runs the same physics in reverse: park a single ion in a known field, measure its cyclotron frequency to a part in $10^{11}$ , and read off its mass — the most precise scale ever built, sharp enough to weigh the energy released in a single nuclear decay. Plasma physics is this article industrialized. A tokamak's design brief is, almost literally, "keep $r_c$ tiny compared to the machine (example 1), close the field lines so nothing escapes along them, and manage the drifts" — with ion-cyclotron-resonance heating broadcasting at $f_c$ to pump the plasma. Above our heads, Earth's dipole field runs the same bottle at planetary scale: Van Allen–belt particles gyrate (milliseconds), bounce between polar mirror points (seconds), and drift around the planet (minutes), a three-clock hierarchy that space-weather models track because the precipitating tail of the loss cone is what paints the aurora and degrades satellite electronics.

Origins · 1 minErnest Orlando Lawrence and how this was discovered

Ernest Orlando Lawrence

1901–1958 · Berkeley

Lawrence found the cyclotron principle in 1929 while browsing a German journal — Wideröe's drift-tube accelerator sketch — and saw that a magnetic field could fold the acceleration path into a spiral, with the speed-independent period (2) keeping a single RF voltage forever in step. The first working model, built with M. Stanley Livingston in 1931, was 4.5 inches across, cost about $25, and reached 80 keV; within a decade Berkeley's 60-inch machine was making medical isotopes and the cyclotron had become the working tool of nuclear physics. Lawrence won the 1939 Nobel Prize, and his lab's response to the relativistic ceiling — McMillan's phase-stability principle, shared independently by Veksler — begat the synchrocyclotron and synchrotron. The laboratory culture he invented, "big science" done by teams around big machines, is as much his legacy as element 103, lawrencium, named for him.

History

1929

— Ernest Lawrence, reading Wideröe's linear-accelerator paper, conceives the magnetic-spiral accelerator resting on the speed-independent period.

1931

— Lawrence and Livingston's first cyclotron, 4.5 inches across, accelerates protons to 80 keV on a fixed RF clock.

1939

— Lawrence receives the Nobel Prize as cyclotrons worldwide reach tens of MeV and the relativistic ceiling comes into view.

1945

— McMillan and Veksler independently find phase stability, unlocking the synchrocyclotron and synchrotron beyond the cyclotron limit.

1952

— Post's magnetic-mirror program turns the adiabatic invariant into a fusion-confinement scheme, loss cone and all.

1958

— Van Allen's Explorer instruments reveal Earth's radiation belts: mirror-trapped particles bouncing between the poles, the bottle at planetary scale.

The physical picture

Picture the magnetic force as a string tethering the particle to a moving center: always perpendicular to the velocity, always pointing toward the middle of an emerging circle, exactly the centripetal pull that uniform circular motion demands. Because the string never pulls forward or backward, the speed is locked and only the direction wheels around. The speed-independence of the period is a precise cancellation, and it pays to see it before deriving it. A particle moving twice as fast needs twice the centripetal force to turn on the same circle — but the magnetic force supplies twice as much, $qvB \propto v$ . The compromise: the faster particle sweeps a circle twice as big, and covers its doubled circumference at doubled speed in the same time. Radius remembers the momentum ( $r_c = p_\perp/qB$ — accelerator physicists call $Br_c = p/q$ the magnetic rigidity); the clock forgets it entirely. Along the field the particle feels nothing — the cross product of parallel vectors vanishes — so it sails freely. A generic launch therefore corkscrews: circle across the field, coast along it.

Figure

The helix: a bead threading a field line — circle across B, free coast along it

Confinement, then, is only two-dimensional so far: the leash binds the particle to a field line but not along it. To plug the ends, make the field stronger there. A gyrating particle behaves like a little current loop with magnetic moment $\mu = mv_\perp^2/2B$ , and slow field changes leave $\mu$ almost exactly constant — so as the particle rides into a high-field throat, $v_\perp^2 = 2\mu B/m$ must grow, and since the total speed cannot change (no work!), $v_\parallel$ must shrink. Squeeze hard enough and $v_\parallel$ hits zero: the particle reflects, as if rolling up a hill. Two such throats make a magnetic bottle.

Figure

The magnetic bottle: strong-field throats reflect gyrating particles

Solving the motion

Everything above must now be earned from the equation of motion. With $\vec E = 0$ and $\vec B$ uniform, the Lorentz law (recalled from the previous articles — we do not re-prove the no-work theorem, we spend it) reads

m\,\frac{\mathrm{d}\vec v}{\mathrm{d}t} = q\,\vec v\times\vec B \qquad (1)

where: $q$ — signed charge; $m$ — mass; $\vec B = B\hat{\vec z}$ once we choose axes. Two conserved quantities are visible before any solving: the speed (magnetic forces do no work), and $v_z$ (the force has no component along $\vec B$ ). Those two facts already cage the answer — constant-speed motion with a frozen axial component can only wheel around in the transverse plane.

Derivation AThe helix, solved from Newton's law

The goal is the complete trajectory of a charge in a uniform field: the gyration frequency, the radius, and the helical shape, from the component equations of (1). The strategy is to expand the cross product, notice that the two transverse velocity components form a coupled linear pair, and decouple them by differentiating — they turn out to execute simple harmonic motion at one special frequency. Watch which quantities cancel on the way; the cancellations are the physics.

Step 1. Expand (1) with $\vec B = B\hat{\vec z}$ . Writing $\omega_c \equiv qB/m$ :

\dot v_x = \omega_c\,v_y, \qquad \dot v_y = -\omega_c\,v_x, \qquad \dot v_z = 0.

Why the split: the $z$ -equation decouples completely — $v_\parallel$ is constant, the free-coast half of the helix, settled at a glance.

Step 2. Decouple the transverse pair by differentiating the first equation and substituting the second:

\ddot v_x = \omega_c\,\dot v_y = -\omega_c^2\,v_x.

Why this is the recognition moment: simple harmonic motion in the velocity components — the velocity vector rotates rigidly at angular frequency $\omega_c$ , which contains the field and the charge-to-mass ratio and nothing else. The speed has already vanished from the clock.

Step 3. Write the transverse solution with $v_\perp$ the constant transverse speed:

v_x = v_\perp\cos(\omega_c t), \qquad v_y = -v_\perp\sin(\omega_c t).

The sense of rotation follows the sign of $q$ : protons and electrons gyrate opposite ways around the same field line. Check the invariant: $v_x^2 + v_y^2 = v_\perp^2$ , constant — consistent with the no-work theorem it must obey.

Step 4. Integrate once more for the position:

x = x_g + \frac{v_\perp}{\omega_c}\sin(\omega_c t), \qquad y = y_g + \frac{v_\perp}{\omega_c}\cos(\omega_c t), \qquad z = z_0 + v_\parallel t.

A circle of radius $r_c = v_\perp/\omega_c = mv_\perp/qB$ about the guiding center $(x_g, y_g)$ , riding along $\vec B$ at $v_\parallel$ — a helix of pitch $v_\parallel T = 2\pi v_\parallel/\omega_c$ .

Step 5. Collect the machine constants:

\omega_c = \frac{qB}{m}, \qquad T = \frac{2\pi m}{qB}, \qquad r_c = \frac{m v_\perp}{qB} = \frac{p_\perp}{qB} \qquad (2)

Why the middle one is the miracle: the period contains no velocity. Every particle of the same $q/m$ in the same field comes around in the same time — slow ones on small circles, fast ones on large. This isochronism is the licence for the fixed-frequency cyclotron below. The last form, $r_c = p_\perp/qB$ , survives relativistically and is the accelerator engineer's rigidity rule. *Verified: substituting back, $\dot v_x = -v_\perp\omega_c\sin\omega_c t = \omega_c v_y$ ✓; results match Griffiths Eqs. 5.7–5.9 and Jackson §12.2. The one-line cross-check: force balance $qv_\perp B = mv_\perp^2/r_c$ gives $r_c = mv_\perp/qB$ and $\omega_c = v_\perp/r_c = qB/m$ — the same constants without solving anything, the speed cancelling in the division.*

Now switch on a uniform electric field across the magnetic one. Intuition from the last article says the particle should accelerate along $\vec E$ — but the magnetic term keeps folding that gained velocity sideways. The result is one of the least intuitive motions in elementary physics: on average the particle moves perpendicular to both fields, at a speed that knows nothing about the particle.

Go deeper · Derivation B · 1 minThe E-cross-B drift of the guiding center

Derivation BThe E-cross-B drift of the guiding center

The goal is the average motion of a charge in uniform crossed fields $\vec E \perp \vec B$ : we will show the gyration continues unchanged around a guiding center that drifts at constant velocity $\vec v_d = \vec E\times\vec B/B^2$ . The strategy is a change of reference velocity — split $\vec v$ into a constant drift plus a remainder and choose the drift to absorb the electric force, so the remainder obeys the pure cyclotron equation just solved. The astonishment to watch for: everything about the particle cancels from the drift.

Step 1. Write the equation of motion with both fields and substitute $\vec v = \vec v_d + \vec v'$ , with $\vec v_d$ constant:

m\,\dot{\vec v}' = q\left(\vec E + \vec v_d\times\vec B\right) + q\,\vec v'\times\vec B.

Step 2. Choose $\vec v_d$ to kill the constant bracket:

\vec E + \vec v_d\times\vec B = 0.

Why we may demand this: it is one vector condition on one unknown vector, and if it can be solved, $\vec v'$ is left obeying exactly equation (1) — pure gyration. The full motion is then "cyclotron circle + steady glide," which is the guiding-center picture.

Step 3. Solve the balance by crossing with $\vec B$ and using the BAC-CAB identity, with $\vec E\perp\vec B$ :

\vec v_d = \frac{\vec E\times\vec B}{B^2} \qquad (3)

Step 4. Interpret. The magnitude is $|\vec v_d| = E/B$ — the velocity-selector balance of the previous article, now recognized as the speed at which the magnetic force on the drifting motion exactly cancels the electric push. And the formula contains no $q$ , no $m$ , no sign: electrons and ions, light and heavy, all drift together, so this drift moves a plasma bodily without driving any current through it. Superposed on the drift, each particle still gyrates; a particle starting from rest traces the classic cycloid — the path of a point on a rolling wheel, Griffiths's Example 5.2 — whose average is exactly $\vec v_d$ . Verified: dimensionally $E/B = (\mathrm{V/m})/\mathrm{T} = \mathrm{m/s}$ ✓; the result matches Chen, Introduction to Plasma Physics §2.2 and Jackson §12.2, and reduces to the velocity-selector condition $v = E/B$ when the gyration amplitude is zero.

Go deeper · Derivation C · 2 minThe magnetic mirror — why strong field reflects particles

Derivation CThe magnetic mirror — why strong field reflects particles

The goal is the confinement mechanism promised by the title: show that a gyrating particle moving into a region of stronger field loses parallel velocity and reflects where $B$ has grown by a factor set only by its launch pitch angle. The strategy has two acts: first identify the gyrating orbit's magnetic moment and state its adiabatic constancy honestly (we verify its consistency rather than prove invariance in full); then let energy conservation do the confining. The trap to avoid: none of this works in a uniform field — it is the field gradient that converts motion between the parallel and perpendicular accounts.

Step 1. Compute the magnetic moment of the gyration, treating the orbit as a current loop. The circulating charge is a current $I = q/T = q\omega_c/2\pi$ enclosing area $A = \pi r_c^2$ :

\mu = IA = \frac{q\omega_c}{2\pi}\,\pi r_c^2 = \frac{q}{2}\,\frac{qB}{m}\,\frac{m^2v_\perp^2}{q^2B^2} = \frac{m v_\perp^2}{2B} \qquad (4)

Why this combination: it is the perpendicular kinetic energy divided by the field strength — a ratio, not an energy, and that is exactly what stays fixed.

Step 2. State the adiabatic invariance. When $B$ changes slowly on the scale of one gyration (in time or along the orbit), $\mu$ is conserved to exponential accuracy — the standard adiabatic-invariant result, which we take from the mechanics of slowly-varying oscillators (proof in Jackson §12.5; the flux $\pi r_c^2 B \propto \mu$ through the orbit is the invariant picture: the orbit clings to a fixed bundle of field lines).

Step 3. Confine with energy conservation. The magnetic force still does no work, so $v^2 = v_\parallel^2 + v_\perp^2$ is constant even in the non-uniform field. As the particle moves into larger $B$ :

v_\perp^2 = \frac{2\mu B}{m} \;\; \text{grows} \qquad\Longrightarrow\qquad v_\parallel^2 = v^2 - \frac{2\mu B}{m} \;\; \text{shrinks}.

The particle reflects — $v_\parallel = 0$ — at the field value $B_{\rm ref} = mv^2/2\mu$ .

Step 4. Derive the loss cone. Launch from the bottle's midplane (field $B_0$ ) at pitch angle $\theta$ to the axis, so $v_\perp = v\sin\theta$ and $\mu = mv^2\sin^2\theta/2B_0$ . Reflection happens before the throat (field $B_{\rm max}$ ) provided $B_{\rm ref} \le B_{\rm max}$ , i.e.

\sin^2\theta \;\ge\; \frac{B_0}{B_{\rm max}} \qquad (5)

Why the bottle always leaks: particles launched too nearly along the axis — inside the "loss cone" $\sin^2\theta < B_0/B_{\rm max}$ — reach the throat with parallel velocity to spare and escape. A mirror ratio $B_{\rm max}/B_0 = 4$ confines only pitch angles above $30^\circ$ ; collisions constantly scatter particles into the cone, which is the fundamental headache of mirror fusion and the reason tokamaks bend the field lines into closed loops instead of plugging ends. *Verified: (4) recomputed by direct substitution ✓; the reflection condition reproduces Chen §2.3.3's mirror criterion and the 30° loss-cone angle for ratio 4 ( $\arcsin\sqrt{1/4} = 30.0^\circ$ ); Earth's belts obey the same criterion with mirror points near the poles.*

The machine: kicks on a speed-independent clock

Bolt the pedal and the steering wheel together and isochronism becomes a technology. Put two hollow D-shaped electrodes ("dees") in a uniform field, apply an oscillating voltage across the gap between them, and inject ions at the center. Inside a dee the ion coasts on its half-circle — shielded from electric fields — and every half-turn it crosses the gap, where the RF voltage, if it has flipped in time, kicks it forward. Equation (2) is the whole trick: every half-turn takes $\pi m/qB$ regardless of speed, so one fixed frequency $f = qB/2\pi m$ stays in step lap after lap while the orbit spirals outward.

Figure

The cyclotron: many small electric kicks, one big magnetic spiral

The exit energy is fixed by the largest circle that fits. At extraction radius $R$ , $v_{\max} = qBR/m$ , so

K_{\max} = \tfrac12 m v_{\max}^2 = \frac{q^2B^2R^2}{2m} \qquad (6)

where: $R$ — dee radius; note the inverse mass — for a given magnet, light particles come out faster but every species obeys the same rigidity budget $BR = p_{\max}/q$ . And note what (6) quietly assumes: that $\omega_c$ stayed put while the energy grew. Relativity breaks that. As $\gamma$ rises the gyration slows to $\omega = qB/\gamma m$ , the ion arrives at the gap later and later relative to the fixed RF, and beyond a few percent of slippage the kicks start landing at the wrong phase. Fixed-frequency cyclotrons therefore top out around tens of MeV for protons; beyond that one must sweep the frequency down as the bunch gains energy (the synchrocyclotron) or ramp $B$ and the RF together to hold one orbit radius (the synchrotron — every GeV and TeV machine on Earth).

Limiting cases

Case	Result	Meaning
$v_\parallel = 0$	flat circle, $r_c = mv_\perp/qB$	the pure cyclotron orbit
$v_\perp = 0$	$\vec F = 0$ , straight line	uniform fields never confine along $\vec B$
general launch	helix, pitch $2\pi v_\parallel/\omega_c$	circle across the field + free coast along it
$v \to c$	$\omega = qB/\gamma m$ , $r_c = p_\perp/qB$	isochronism lost — synchrotrons take over; rigidity form survives
add $\vec E \perp \vec B$	guiding center drifts at $\vec E\times\vec B/B^2$	charge- and mass-blind; cycloid for a particle from rest
slowly varying $B$	$\mu = mv_\perp^2/2B$ conserved	mirrors, bottles, Van Allen belts; loss cone $\sin^2\theta = B_0/B_{\rm max}$
$q \to -q$	opposite sense of gyration	electrons and ions counter-rotate; drift (3) unchanged

Common trapThe cyclotron frequency is speed-independent — until it very much is not

$\omega_c = qB/m$ contains no velocity, and the whole classical cyclotron leans on that. But the exact statement is $\omega = qB/\gamma m$ : what is speed-independent is only the non-relativistic clock. At $\gamma = 1.03$ — a 27 MeV proton — the orbit has already slipped 3% against a fixed RF, and the kicks begin to miss. Related traps: the radius is not speed-independent (it is the momentum meter, $r_c = p_\perp/qB$ ); the drift $\vec E\times\vec B/B^2$ is a guiding-center average, not the instantaneous velocity (the particle still gyrates, and one starting from rest traces a cycloid, momentarily stopping once per cycle); and $\mu$ -conservation is adiabatic, holding only while $B$ changes slowly over a gyration — slam the field on suddenly and the mirror force vanishes with the invariance.

Worked examples

1 · A fusion deuteron on its leash (confidence builder)

In a magnetic-confinement experiment, a deuteron ( $m = 3.34\times10^{-27}\ \mathrm{kg}$ , $q = e$ ) has kinetic energy $10\ \mathrm{keV}$ , moving perpendicular to a $B = 5.0\ \mathrm{T}$ field. Find its speed, gyroradius, and cyclotron frequency.

Step 1. Speed from the kinetic energy: $v = \sqrt{2K/m} = \sqrt{2(10^4)(1.60\times10^{-19})/(3.34\times10^{-27})} = 9.8\times10^{5}\ \mathrm{m/s}$ — 0.3% of $c$ , comfortably non-relativistic.

Step 2. Gyroradius from (2): $r_c = \dfrac{mv}{qB} = \dfrac{(3.34\times10^{-27})(9.8\times10^{5})}{(1.60\times10^{-19})(5.0)} = \boxed{4.1\ \mathrm{mm}}$ — the deuteron is tied to its field line on a millimetre leash inside a metres-wide plasma; this scale separation is what makes "particles follow field lines" a working approximation in fusion design.

Step 3. Cyclotron frequency: $f_c = \dfrac{qB}{2\pi m} = \dfrac{(1.60\times10^{-19})(5.0)}{2\pi(3.34\times10^{-27})} = \boxed{38\ \mathrm{MHz}}$ — radio frequency, and indeed "ion cyclotron resonance heating" pumps power into tokamak plasmas by broadcasting at exactly this frequency.

Step 4. Contrast the electron: same energy, mass smaller by $\sim 3670$ , so $v$ is 60× larger but $r_c = mv/qB \propto \sqrt{m}$ shrinks to $0.07\ \mathrm{mm}$ — electrons hug field lines even tighter, gyrating in the GHz.

2 · Design numbers for a proton cyclotron (exam level)

A cyclotron uses $B = 1.5\ \mathrm{T}$ and has dees of radius $R = 0.50\ \mathrm{m}$ . For protons ( $m = 1.67\times10^{-27}\ \mathrm{kg}$ ): (a) what RF frequency keeps the kicks in step? (b) what is the maximum kinetic energy delivered? (c) should the designer worry about relativity?

Step 1. The RF must match the cyclotron frequency (2): $f_c = \dfrac{qB}{2\pi m} = \dfrac{(1.60\times10^{-19})(1.5)}{2\pi(1.67\times10^{-27})} = \boxed{23\ \mathrm{MHz}}$ — fixed, because the period ignores the speed.

Step 2. The exit speed is set by the largest orbit that fits: $v_{\max} = \dfrac{qBR}{m} = \dfrac{(1.60\times10^{-19})(1.5)(0.50)}{1.67\times10^{-27}} = 7.2\times10^{7}\ \mathrm{m/s}$ .

Step 3. Maximum kinetic energy from (6): $K_{\max} = \dfrac{q^2B^2R^2}{2m} = \tfrac12(1.67\times10^{-27})(7.2\times10^{7})^2 = 4.3\times10^{-12}\ \mathrm{J}$ .

Step 4. Convert to physicist's units: $K_{\max} = 4.3\times10^{-12}/1.60\times10^{-13} = \boxed{27\ \mathrm{MeV}}$ — squarely in the range of the isotope-production cyclotrons in hospital basements today.

Step 5. Relativity check: $v_{\max}/c = 0.24$ , so $\gamma = 1/\sqrt{1 - 0.24^2} = 1.03$ . The orbital frequency has already slipped $3\%$ below the RF by exit — tolerable here with phase margin, fatal at higher energy. That slippage is exactly why GeV-scale machines are synchrotrons, ramping $B$ and RF together as $\gamma$ grows.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.1.2–5.1.3 (cyclotron motion, cycloid) · Jackson, Classical Electrodynamics, 3rd ed., §12.2 and §12.5 (uniform-field motion; adiabatic invariants) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6 · Zangwill, Modern Electrodynamics, §12.2 · Chen, Introduction to Plasma Physics and Controlled Fusion, Ch. 2.

Provenance. All equations standard textbook results. A verifier should: (1) confirm $(\mathbf{v}\times\mathbf{B})\cdot\mathbf{v}=0$ giving energy conservation; (2) re-solve $m\dot{\mathbf{v}}=q\mathbf{v}\times\mathbf{B}$ for $\mathbf{B}=B\hat z$ to recover $\omega_c=qB/m$ , $r_c=mv_\perp/(qB)$ (Griffiths §5.1.2 Eqs. 5.7-5.9; Jackson §12.2); (3) verify $\mu=IA=mv_\perp^2/(2B)$ ; (4) check the $\mathbf{E}\times\mathbf{B}$ drift $\mathbf{v}_d=\mathbf{E}\times\mathbf{B}/B^2$ (Chen Ch. 2). No empirical constants quoted; relativistic generalization $\omega_c=qB/(\gamma m)$ standard. Worked numbers recomputed for this edition: deuteron 10 keV in 5 T — $v = 9.8\times10^5$ m/s, $r_c = 4.1$ mm, $f_c = 38$ MHz (electron at same energy: $r_c = 0.07$ mm); proton cyclotron 1.5 T, 0.50 m — $f_c = 23$ MHz, $v_{\max} = 7.2\times10^7$ m/s, $K_{\max} = 27$ MeV by both $\tfrac12mv^2$ and $q^2B^2R^2/2m$ , $\gamma = 1.03$ ; loss cone for mirror ratio 4 = $30.0^\circ$ .

Builds on: Lorentz Force Law · Magnetic Force on Moving Charges

Leads to: Electric Current and Current Density · Biot-Savart Law

If the period is speed-independent, why does the radius change at all — couldn't the particle just go faster on the same circle?

Newton forbids it. On a circle of radius $r$ at speed $v$ the required centripetal force is $mv^2/r$ , growing like $v^2$ , while the magnetic force only grows like $v$ ( $qvB$ ). A particle kicked to higher speed on its old circle is under-steered, so it swings wider until the balance $qvB = mv^2/r$ is restored at $r = mv/qB$ . The period stays fixed because the radius absorbs the speed change — bigger circle, faster lap, same time. Radius is the momentum meter; the clock is the $q/m$ meter.

Do electrons and protons in the same field orbit in the same direction?

No — the sign of $q$ sets the sense of rotation, so they counter-rotate around the same field line (an electron's $\omega_c$ vector points along $\vec B$ , a proton's against it, in the standard convention). But the crossed-field drift $\vec E\times\vec B/B^2$ is the same for both in magnitude and direction — the sign flips twice, once in the force and once in the gyration sense. That is why the drift moves a whole plasma sideways without separating its charges or driving a current.

Where does the energy come from in a cyclotron if the magnetic field does no work?

Entirely from the RF electric field in the gap — the magnetic field is the steering, never the pedal (the division of labour proven in Lorentz Force Law). Each gap crossing adds $q\Delta V$ ; the field between the dees only folds the path back for another kick. Count the bookkeeping in worked example 2: 27 MeV out with, say, 50 kV per crossing means roughly 270 full turns — 540 electric kicks, zero magnetic joules. The magnet's contribution is geometric: it makes a metre-long machine do the work of a half-kilometre linac.

Is the magnetic moment mu really conserved, or only approximately?

Approximately, but extremely well. $\mu$ is an adiabatic invariant: when the field seen by the particle changes slowly compared with one gyration, the change in $\mu$ per bounce is exponentially small in the slowness ratio — not merely first-order small. It is not conserved when the field changes abruptly (a shock front, a sharp field kink, gyration-scale turbulence), and such violations are physically important: they scatter trapped particles into the loss cone and let radiation-belt particles precipitate. The rigorous machinery (action integrals, $J = \oint p\,\mathrm{d}q$ ) lives in advanced mechanics; the working rule is "slow change → $\mu$ fixed → mirror force real."

Why do fusion devices prefer tokamaks over magnetic mirrors if mirrors follow so directly from mu-conservation?

The loss cone never closes. Equation (5) says a mirror confines only particles with pitch angles above $\arcsin\sqrt{B_0/B_{\rm max}}$ , and Coulomb collisions relentlessly scatter particles into the cone, draining the bottle on a collisional timescale. Fusion plasmas are collisional enough that simple mirrors leak fatally (decades of clever fixes — minimum-B baseball coils, tandem mirrors — narrowed but never closed the gap). The tokamak's answer is topological: bend the field lines into closed nested tori so there is no "end" to leak from, then spend the engineering budget on the drifts and instabilities that curved fields introduce instead.

The E×B drift speed is E/B — the same expression as the velocity selector's pass speed. Coincidence?

Same physics, two costumes. The selector passes particles already moving at $v = E/B$ because at that velocity the magnetic force exactly cancels the electric one — zero net force, straight line. The drift result says: whatever the particle's state, the average motion settles at exactly that same velocity, because it is the unique frame in which the electric field's push is absorbed by the magnetic force and the leftover dynamics is pure gyration. A selector particle is just one whose gyration amplitude happens to be zero. Transform to the frame moving at $\vec E\times\vec B/B^2$ and the electric field vanishes — a first taste of how $\vec E$ and $\vec B$ mix under boosts.

Does gyrating radiate? The particle is accelerating constantly.

Yes — centripetal acceleration is acceleration, and a gyrating charge emits cyclotron radiation (called synchrotron radiation when relativistic), slowly draining $v_\perp$ at fixed $\mu$ -per-energy bookkeeping. For heavy, slow particles the loss is negligible (a 10 keV deuteron radiates immeasurably little over any laboratory timescale), which is why this article could treat orbits as eternal. For light or fast particles it dominates the engineering: synchrotron losses grow like $\gamma^4/r^2$ , are the reason electron accelerators above ~100 GeV must be linear, and — turned from bug to feature — are the brilliant X-ray light sources that synchrotron facilities sell beamtime on. The radiation story is owned by this track's radiation tier.

Cyclotron Motion and Magnetic Confinement

Where this lives today

The physical picture

Solving the motion

The machine: kicks on a speed-independent clock

Limiting cases

Worked examples

1 · A fusion deuteron on its leash (confidence builder)

2 · Design numbers for a proton cyclotron (exam level)

Going further

Before this

Where this leads