CyclotronJEE Advanced

Spiralling particles, constant frequency — interactive Physics simulation for IIT-JEE.

Concept

A cyclotron accelerates ions with a fixed-frequency alternating voltage across the gap between two D-shaped electrodes in a magnetic field. The magic: the orbital period $2\pi m/qB$ is independent of speed — faster particles trace bigger circles in exactly the same time, so the same RF clock kicks them every half-turn as they spiral outward.

Key formula

f = \frac{qB}{2\pi m}, \qquad r = \frac{mv}{qB}, \qquad KE_{max} = \frac{q^2B^2R^2}{2m}

Derivation

Magnetic force supplies centripetal force: $qvB = mv^2/r \Rightarrow r = mv/qB$ .

Period $T = 2\pi r/v = 2\pi m/qB$ — v cancels! Each gap crossing adds $qV$ of energy; the radius grows as $\sqrt{KE}$ until the particle exits at $R$ with $v = qBR/m$ .

Scenarios to explore

Cyclotron — Spiral accelerator — same period every turn, KE = q²B²R²/2m.

Real-world applications

Medical isotope production (PET tracers like ¹⁸F).
Proton therapy for tumours.
First artificial nuclear reactions (Lawrence, 1932).

JEE exam tips

KE_max = q²B²R²/2m: independent of dee voltage — a favourite trick question.
Relativity breaks the trick: as m grows near c, f drifts (fix = synchrocyclotron).
Electrons are useless in cyclotrons — they go relativistic almost immediately.

Common mistakes

Thinking the magnetic field does work — it never does; the ELECTRIC field in the gap accelerates.
Making the RF frequency depend on speed — constancy of f is the whole point.
KE_max depends on V — it doesn't (V only changes how many turns it takes).

Exam traps to avoid

The particle gains energy TWICE per revolution (two gap crossings).
Inside the dees the speed is constant — energy changes only in the gap.