JEE/NEET Physics · Moving Charges & Magnetism series · Part 3 of 8 · All parts →
- Crossed E and B fields: only ONE speed passes straight — v = E/B
- Velocity selector: filters a beam to a single speed before analysis
- Cyclotron: alternating voltage + fixed B spirals particles to high energy
- Cyclotron frequency f = qB/(2πm) — constant, so the timing never needs adjusting
- Limit: at relativistic speeds, timing breaks — synchrocyclotrons take over
Cross an electric field with a magnetic one and you build a filter that only one speed of particle can survive. Chain that trick with the speed-blind circle of Part 2, and you can push particles to millions of volts of energy with a modest power supply. Part 3 of the Moving Charges & Magnetism series.
- The crossed-field gate
- The velocity selector
- The cyclotron’s spiral ladder
- Why the timing never changes
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Crossed-Field Gate
Send a positive charge through E (pushing down, say) and B (pushing up, for that geometry). Electric force qE is speed-blind; magnetic qvB grows with speed. They cancel for exactly one speed:
The Velocity Selector
Place slits at both ends: every speed except E/B gets deflected into the walls. Out comes a mono-energetic beam — the essential preparation for mass spectrometers and precision experiments.
The Cyclotron’s Spiral Ladder
Two semicircular ‘dees’ sit in a strong B; an alternating voltage flips across the gap. Each gap crossing kicks the particle’s speed up; each dee visit turns it in a half-circle. Speed grows → radius grows (r = mv/qB) → an outward spiral of ever-bigger semicircles, gaining energy per gap-crossing until extracted.
Why the Timing Never Changes
Because faster particles ride wider circles in the same time, the voltage’s flips never need retuning — the particle is ALWAYS at the gap exactly when the voltage is right. This coincidence (from Part 2) is the cyclotron’s licence to exist.
Solved Examples
v = E/B = 6×10⁵ m/s.
✔
Answer: 6×10⁵ m/s
f = qB/(2πm) = (1.6×10⁻¹⁹ × 2)/(2π × 1.67×10⁻²⁷) ≈ 3×10⁷ Hz — 30 MHz.
✔
Answer: ≈30 MHz
v_max = qBR/m = (1.6×10⁻¹⁹ × 1.5 × 0.5)/1.67×10⁻²⁷ ≈ 7.2×10⁷ m/s.
K = ½mv² ≈ 4.3 MeV (nonrelativistically valid — barely).
✔
Answer: ≈4.3 MeV
- Selector passing any charge. The gate selects by SPEED only — mass and sign don’t matter (both forces flip together with sign).
- Thinking the dees’ voltage does the circling. B does the turning; the gap voltage only boosts speed — division of labour.
- Retuning frequency as speed grows. Never — that speed-independence is the machine’s whole trick (until relativity breaks it).
- Ignoring the relativistic limit. Near light speed, mass effectively grows and timing slips: cyclotrons stall; synchrotrons retune.
This Physics in Your Daily Life
- Mass spectrometers in labs and airports — selector gate + circle radius = molecule identification: security screening is this card in a box.
- Hospital proton-therapy cyclotrons — 230 MeV proton beams stopping precisely in tumours: cancer treatment as spiral engineering.
- Radioisotope production for PET scans — medical cyclotrons brew short-lived tracers daily: hospital basements hosting Part 3.
- The LHC’s ancestry — Lawrence’s 1930 palm-sized cyclotron grew into the 27 km synchrotron: same idea, relativistic version.
- Older TV and oscilloscopes — used related crossed-field steering to aim beams: the family tree on every desk until flat screens.
Two doormen with opposite styles: one (electric) shoves everyone equally; the other (magnetic) shoves in proportion to how fast you walk. For one walking speed their shoves cancel exactly — walk slower and the electric doorman wins (deflected one way); faster and the magnetic doorman wins (deflected the other). One speed walks through untouched.
E = 10⁵ V/m, B = 0.1 T: only 10⁶ m/s passes. Halve the speed? Magnetic push halves, electric stays: net electric deflection. Double it? Magnetic doubles: deflected backwards. The gate’s selectivity is absolute — a velocity knife sharpened by geometry.
Draw the channel: E arrows down, B-into-page crosses, velocity rightward. Electric force arrow down (fixed length); magnetic arrow up (grows with v). At exactly one v, the two arrows match lengths — the drawing balances, the particle sails. The balanced diagram IS v = E/B.
Practice set (answers hidden — try first)
(NEET-level) E = 2×10⁵, B = 0.4: selected v =
(JEE Main-level) The velocity selector’s passed speed depends on
(NEET-level) In a cyclotron, particles gain energy
(Concept) Cyclotron frequency is independent of
(JEE Main-level) Doubling B in a cyclotron: f
- v = E/B: the one speed that passes
- both force signs flip together: gate is charge-blind
- cyclotron = B-circle + gap kicks
- f = qB/2πm, fixed forever
- relativistic mass growth stalls it
- 🔁 selector condition and logic
- 🔁 cyclotron operation
- 🔁 frequency formula
- 🧠 Chant: ‘E over B decides who walks through’.
- 🧠 Cyclotron: ‘fixed tune, growing circles’.
- 🏠 Daily: airport chemical scanners = selector + circle.
- 🏠 Daily: proton therapy = spiral ladder in hospital basements.
Quick revision
- Crossed E and B fields: only ONE speed passes straight — v = E/B
- Velocity selector: filters a beam to a single speed before analysis
- Cyclotron: alternating voltage + fixed B spirals particles to high energy
- Cyclotron frequency f = qB/(2πm) — constant, so the timing never needs adjusting
- Limit: at relativistic speeds, timing breaks — synchrocyclotrons take over
- The cyclotron’s spiral ladder
Have a doubt on this topic?
