Editing Cyclotron (section)

=== Particle trajectory ===
[[File:Spiral-fermat-1.svg|thumb|250px|The trajectory followed by a particle in the cyclotron approximated with a [[Fermat's spiral]]]]

While the trajectory followed by a particle in the cyclotron is conventionally referred to as a "spiral", it is more accurately described as a series of arcs of constant radius. The particles' speed, and therefore orbital radius, only increases at the accelerating gaps. Away from those regions, the particle will orbit (to a first approximation) at a fixed radius.<ref name="Chautard">{{cite journal |last1=Chautard |first1=F |title=Beam dynamics for cyclotrons |journal=CERN Particle Accelerator School |date=2006 |pages=209–229 |doi=10.5170/CERN-2006-012.209 |url=https://cds.cern.ch/record/1005052/files/p209.pdf |access-date=4 July 2022}}</ref>

Assuming a uniform energy gain per orbit (which is only valid in the non-relativistic case), the average orbit may be approximated by a simple spiral. If the energy gain per turn is given by {{math|&Delta;{{var|E}}}}, the particle energy after {{mvar|n}} turns will be:
<math display="block">E(n) = n \Delta E</math>
Combining this with the non-relativistic equation for the kinetic energy of a particle in a cyclotron gives:
<math display="block">r(n) = {\sqrt{2 m \Delta E} \over q B} \sqrt{n}</math>
This is the equation of a [[Fermat's spiral|Fermat spiral]].