Tip: In pure B mode, B does no work — speed stays constant, only direction changes. Energy is conserved in cyclotron motion.
Section 02
The Idea, Step by Step
A magnet near a moving electric charge gives it a sideways shove — never a forward push, never a backward push, always a push at right angles to the way it's going. Think of whirling a ball on a string: the string pulls toward the center, so the ball loops in a circle instead of speeding up. A magnetic field plays exactly that trick on a moving charge, bending its straight path into a circle while leaving its speed untouched.
To put numbers on it, you need three things: the charge $q$, how fast it's going $v$, and how strong the field is $B$. When the motion is square-on to the field, the sideways force is simply $F = qvB$. Take an electron ($q = 1.6\times10^{-19}$ C) zipping at $v = 5\times10^{5}$ m/s through a field of $B = 0.2$ T: the force is $F = (1.6\times10^{-19})(5\times10^{5})(0.2) \approx 1.6\times10^{-14}$ N. That sounds vanishingly small, but the electron is far tinier still, so it whips around a circle thousands of times a second. Setting this sideways force equal to the circular-motion requirement $mv^2/r$ gives the size of that circle:
A faster particle makes a bigger circle but travels proportionally faster, so the time for one lap $T_c$ comes out the same — that's the secret of the cyclotron.
The complete law is $\mathbf F = q\mathbf E + q\,\mathbf v \times \mathbf B$. The cross product is the key: the magnetic part always points perpendicular to $\mathbf v$, so it does zero work — kinetic energy and speed never change, only direction. Because $T_c = 2\pi m/(qB)$ doesn't contain $v$, every orbit takes the same time regardless of energy (this isochronism is why a fixed-frequency cyclotron stays in step with the particle as it spirals out). And if the velocity also has a slice pointing along $\mathbf B$, that slice feels no force and sails straight through, so the circle stretches into a helix. On the sliders: $B$ sets field strength (bigger $B$ → tighter, quicker orbit), $q/m$ picks the particle, $v_0$ sets the speed, and $v_\parallel/v$ controls how stretched the helix is.
Try this in the sim above: (1) Slide $B$ up and watch $r$ shrink while $f_c$ and $T_c$ stay locked to $q/m$ — speed-independent, just as the formula promises. (2) Switch to E×B Drift, turn on $E$, and see the whole circle slide sideways at $v = E/B$ — the same drift for any charge. (3) In Helical View, push $v_\parallel/v$ from 0 up to 1 and watch the flat circle wind into a corkscrew, then straighten into a line.
Section 03
Equation Derivation
A charged particle moving in electric and magnetic fields experiences the Lorentz force. In a uniform magnetic field alone, the particle traces a perfect circle — the basis of cyclotrons, mass spectrometers, and the aurora.
Governing Equation — Lorentz Force
$$\boxed{\;\mathbf F = q\mathbf E + q\,\mathbf v \times \mathbf B\;}$$
$$r = \frac{m v_\perp}{|q|B}, \qquad \omega_c = \frac{|q|B}{m}, \qquad T_c = \frac{2\pi m}{|q|B}$$
Symbol
Meaning
SI Unit
$\mathbf F$
Force on the charge
N
$q$
Particle charge (signed)
C
$\mathbf E$
Electric field
V/m
$\mathbf B$
Magnetic field
T
$\mathbf v$
Particle velocity
m/s
$m$
Particle mass
kg
$r$
Cyclotron (gyro) radius
m
$\omega_c$
Cyclotron angular frequency
rad/s
$f_c$
Cyclotron frequency = $\omega_c/2\pi$
Hz
$v_\perp$
Speed perpendicular to B
m/s
Step 1 — The magnetic force is always perpendicular to v
From $\mathbf F = q\,\mathbf v \times \mathbf B$, the force is perpendicular to both $\mathbf v$ and $\mathbf B$. Power delivered:
$$P = \mathbf F \cdot \mathbf v = q(\mathbf v \times \mathbf B)\cdot\mathbf v = 0$$
A magnetic field does no work. Kinetic energy stays constant; only the direction of $\mathbf v$ changes.
Step 2 — Circular motion in a uniform B-field
For $\mathbf v \perp \mathbf B$, the magnetic force has constant magnitude $|q|vB$ — exactly a centripetal force:
Angular frequency $\omega_c = v/r = |q|B/m$ — independent of $v$. All particles complete one orbit in the same time $T_c = 2\pi m/(|q|B)$.
This is the principle of the cyclotron: an alternating voltage at exactly $\omega_c$ accelerates the particle each gap-crossing, growing $r$ but keeping $T_c$ fixed.
Step 3 — Helical motion when $v_\parallel \ne 0$
Decompose $\mathbf v = \mathbf v_\parallel + \mathbf v_\perp$ along $\mathbf B$. Parallel component feels no force; perpendicular component circles. Together: a helix.
$$\text{Pitch} = v_\parallel \cdot T_c = \frac{2\pi m v_\parallel}{|q|B}$$
This is how cosmic rays spiral along Earth's magnetic field lines, dumping energy near the poles to create the aurora.
Step 4 — Crossed E and B: the drift velocity
For uniform $\mathbf E \perp \mathbf B$, the steady-state motion is a circle plus a constant drift:
$$\boxed{\;\mathbf v_{\text{drift}} = \frac{\mathbf E \times \mathbf B}{B^2}\;}$$
Crucially, $\mathbf v_{\text{drift}}$ is independent of charge sign and mass — both ions and electrons drift together. This is the basis of velocity selectors: only $v = E/B$ passes straight through.
Step 5 — Mass spectrometer (Bainbridge, 1933)
After velocity selection (all ions have $v$), they enter pure B. Each ion traces a semicircle of radius $r = mv/(qB)$. Different mass → different radius:
$$\frac{m}{q} = \frac{r B}{v}$$
Modern mass spectrometers measure $m/q$ to a few parts per billion — identifying isotopes, peptides, and complex molecules.
Mapping to the simulation
• Slider B sets magnetic field — increasing B tightens the orbit and speeds up rotation.
• Slider E turns on a perpendicular E-field for E×B drift.
• Slider v∥/v sets the helix pitch — 0 = pure circle, 1 = straight-line along B.
• Mass-spectrometer mode shows three particles of different m/q diverging into different radius circles.
Reference: Griffiths — Introduction to Electrodynamics, 4th Ed., Pearson, §5.1.2: "Magnetic Forces"; HRW 10th Ed., §28-1 to §28-7: "Magnetic Fields"; Jackson — Classical Electrodynamics, 3rd Ed., §12.2: "Motion in a Uniform, Static Magnetic Field".
Section 04
Frequently Asked Questions
From the radius formula $r = mv/(qB)$, faster particles trace bigger circles. The period to go around a circle is $T = 2\pi r/v = 2\pi m/(qB)$ — the $v$ cancels exactly. So a slow electron and a fast electron in the same magnetic field both complete one full loop in the same time. This is what makes the cyclotron work: a fixed-frequency RF voltage stays in resonance with the particle as it accelerates, gradually spiralling outward.
Key: Bigger orbit at higher speed, but proportionally faster motion → same period. This isochronism is the cyclotron's defining property.
Mass spectrometers in chemistry labs and crime forensics. Cyclotrons and synchrotrons (LHC, RHIC) accelerate particles for high-energy physics. Cathode-ray tubes (old TVs and oscilloscopes) deflect electron beams using B-fields. Aurora — solar-wind ions spiral along Earth's magnetic field lines and dump energy at the poles. Tokamaks confine fusion plasma magnetically. Hall-effect sensors detect magnetic fields in industry. MRI machines use Lorentz force on nuclear spins. Even simple electric motors and generators are Lorentz force at work — current carries charges, B exerts force, motor turns.
Key: From your phone's electric motor to the LHC, every charged-particle device runs on Lorentz force.
In Cyclotron mode you see a charged particle (red dot) starting with velocity $v_0$ in a field $\mathbf B$ pointing out of the page (×-symbol grid). The Lorentz force $q\mathbf v\times\mathbf B$ continuously rotates v, producing a perfect circle. The particle's trail traces the orbit. In E×B drift mode, an additional E-field perpendicular to B causes the centre of the circle itself to drift sideways — a cycloid. In helical mode you see a 3D-projected helix. In mass spec mode, three particles of different m/q simultaneously enter at the same speed but trace different-radius semicircles.
Key: Watch the speed readout — in pure-B modes it's exactly constant. The radius readout matches $r = mv/(qB)$ in real time.
Work is $W = \int \mathbf F\cdot d\mathbf s = \int \mathbf F\cdot\mathbf v\,dt$. For magnetic force, $\mathbf F = q\mathbf v\times\mathbf B$, so $\mathbf F\cdot\mathbf v = q(\mathbf v\times\mathbf B)\cdot\mathbf v$. The cross product $\mathbf v\times\mathbf B$ is perpendicular to $\mathbf v$ by definition, so the dot product is exactly zero. Force does no work → kinetic energy is constant → speed is constant. Only the direction of motion changes. This is why magnets can deflect beams but never speed them up — accelerating particles requires E-fields (which is why cyclotrons need an alternating voltage gap, not just a magnet).
Key: Magnetic forces steer; electric forces accelerate. Together they shape every charged-particle device.
At relativistic speeds, the inertial mass becomes $m = \gamma m_0$ where $\gamma = 1/\sqrt{1-v^2/c^2}$. The cyclotron frequency $\omega_c = qB/(\gamma m_0)$ now decreases with energy. A fixed-frequency RF voltage falls out of phase with the particle, halting acceleration. Solutions: (1) synchrocyclotrons reduce RF frequency over each spiral; (2) isochronous cyclotrons use spatially varying B to compensate; (3) synchrotrons ramp B and RF frequency together while keeping the radius fixed (think LHC: 27 km ring). The LHC's 7 TeV protons have $\gamma \approx 7500$ — completely off-scale for a classical cyclotron.
Key: Relativity breaks the cyclotron's isochronism. Synchrotrons solve it by adjusting B and RF dynamically.
In the rest frame moving at $\mathbf v_d = \mathbf E\times\mathbf B/B^2$, the electric field transforms away (Lorentz boost: $\mathbf E' = \mathbf E + \mathbf v_d\times\mathbf B = 0$ for this specific $\mathbf v_d$). In that frame, only B remains — the particle just circles. Back in the lab frame, you see a circle plus a uniform drift. Since the drift transformation depends only on E and B (not on the particle's properties), it's identical for all charges and masses. This is why fusion plasmas have ions and electrons drifting together rather than separating — a crucial property for plasma confinement.
Key: E×B drift is a frame-transformation effect — it must be charge-and-mass-independent.
The Sun ejects high-energy charged particles (mostly protons and electrons) in the solar wind. When they reach Earth's magnetosphere, the Lorentz force $q\mathbf v\times\mathbf B$ traps them on helical paths along magnetic field lines. Field lines converge near the magnetic poles, intensifying B and tightening the helix; at sufficiently strong B the magnetic mirror effect reflects most particles back. Some "leak through" the mirror at the poles, plunging into the upper atmosphere. There they collide with $\mathrm O$ and $\mathrm N_2$ molecules, exciting them; the molecules then de-excite, emitting characteristic light: green/red from oxygen, blue/purple from nitrogen. The Lorentz force is what funnels the particles to the polar regions in the first place.
Key: Aurora = Lorentz force funnel + atomic emission spectra. The whole show is electromagnetic physics on a planetary scale.
Best Resource: HyperPhysics — "Charged Particle in a Magnetic Field" & "Cyclotron"; Khan Academy — "Magnetic forces and magnetic fields"; MIT OCW 8.02 Electricity and Magnetism, Lectures 17-18; PhET Simulations — "Charges and Fields".
Section 05
Common Misconceptions
❌ A magnetic field accelerates charged particles to high energies.
✅ Magnetic fields only deflect — they never change kinetic energy.
A common confusion stemming from cyclotrons "accelerating" particles. The acceleration in a cyclotron actually comes from the alternating electric field across the dee gap; the magnetic field's only job is to bend the trajectory back so the particle re-crosses that gap repeatedly. In a pure uniform magnetic field, a particle's speed is constant forever — only its direction changes. Mathematically: $\mathbf F\cdot\mathbf v = 0$ for magnetic force, so power delivered is zero.
📖 HRW 10th Ed., §28-3: "Crossed Fields: Discovery of the Electron"; Griffiths §5.1.
❌ The cyclotron frequency depends on the particle's speed.
✅ For non-relativistic particles, $\omega_c = qB/m$ depends only on $q/m$ and $B$.
A faster particle traces a bigger circle but at proportionally higher speed, so it completes one orbit in the same time. This isochronism is what allows cyclotrons to operate with a fixed RF frequency. At relativistic speeds the effective mass $\gamma m$ grows with energy, breaking isochronism — that's why high-energy machines need synchrocyclotrons (variable frequency) or isochronous cyclotrons (radially shaped B-field).
📖 Griffiths — Introduction to Electrodynamics, 4th Ed., §5.1.2; Jackson §12.2.
❌ A positive charge always orbits clockwise in a magnetic field.
✅ Direction depends on the orientation of B (out-of-page vs into-page) and the sign of q.
Use the right-hand rule for $\mathbf v\times\mathbf B$. For B out of the page and positive q, the force on a particle moving rightward is downward — the particle circles clockwise. Reverse B (into page) and it goes counter-clockwise. Reverse q (electron) and the force flips. The simulation makes this explicit: switch q-sign mentally and watch the orbit direction reverse.
📖 HRW 10th Ed., §28-2: "The Definition of Magnetic Field"; Serway 8th Ed., §29.1.
❌ E×B drift depends on the particle's charge — positive and negative drift opposite ways.
✅ Both signs drift in the same direction at the same speed.
This surprises everyone the first time. The drift velocity $\mathbf v_d = \mathbf E\times\mathbf B/B^2$ is a frame-transformation effect, not a charge-dependent kinematic effect. Positive and negative ions both drift together with the same velocity. This is essential for plasma confinement: if ions and electrons drifted oppositely, charge separation would build huge electric fields and rip plasmas apart. Because they drift together, the plasma maintains quasi-neutrality.
📖 Chen — Introduction to Plasma Physics and Controlled Fusion, 3rd Ed., §2.2.
❌ A charged particle moving parallel to B feels no force at all.
✅ It feels no magnetic force, but any E-field still acts on it.
The cross product $\mathbf v\times\mathbf B = 0$ when $\mathbf v\parallel\mathbf B$, so the magnetic part of the Lorentz force vanishes. But the full Lorentz force is $q\mathbf E + q\mathbf v\times\mathbf B$ — any electric field still produces a force. In a real device with both E and B fields, parallel-moving particles still feel $q\mathbf E$. This becomes important in plasma physics where particle "guiding centres" can drift along field lines under E-field acceleration even when the perpendicular gyromotion is decoupled from E.
❌ The cyclotron radius $r = mv/(qB)$ applies to any particle in any magnetic field.
✅ It's only exact for uniform B, non-relativistic motion, and pure perpendicular velocity.
In non-uniform fields the radius is only approximate (locally), and additional drifts appear (gradient-B drift, curvature drift). At relativistic speeds, replace $m$ with $\gamma m_0$, giving $r = p_\perp/(|q|B)$ in terms of relativistic momentum. With non-perpendicular v, only $v_\perp$ enters the radius; $v_\parallel$ produces helical pitch. So the simple formula is the textbook ideal — real-world charged-particle motion in space and lab plasmas needs the full machinery.
📖 Boyd & Sanderson — The Physics of Plasmas, Cambridge UP, 2003, Ch. 2.
Misconception research: Maloney, O'Kuma, Hieggelke & Van Heuvelen — "Surveying students' conceptual knowledge of electricity and magnetism", Am. J. Phys. 69, S12 (2001); Saglam & Millar — "Upper high school students' understanding of electromagnetism", Int. J. Sci. Educ. 28, 543 (2006); Guisasola et al. — "Difficulties in learning the introductory magnetic field theory", Sci. Educ. 88, 443 (2004).