For perpendicular entry into a uniform magnetic field, ∣q∣vB=rmv2 gives r=(∣q∣B)mv. The radius increases with momentum and decreases with field strength or charge magnitude. Combining with T=v2πr gives T=(∣q∣B)2πm, independent of speed in the nonrelativistic model. With a known accelerating voltage V,mq can be inferred from r and B using m∣q∣=(B2r2)2V.
A cyclotron uses magnetic fields inside its dees to bend ions, while an alternating electric field across the gap adds energy. The electric frequency matches the orbital frequency. At maximum orbit radius r,K=q2B22mr2. Increasing gap voltage changes the energy added per crossing, but not this ideal exit energy at fixed radius and field. Relativistic mass-energy behaviour increases the period at high speed, disrupting fixed-frequency synchronisation.
Example 1
An ion has m=4.00×10−27kg,∣q∣=2.00×10−19C,v=1.00×106ms−1 and B=0.500T.
r=(∣q∣B)mv=0.0400m.
Example 2
A cyclotron ion has m=4.00×10−27kg,∣q∣=2.00×10−19C,B=0.500T and exit r=0.100m.
K=q2B22mr2=1.25×10−14J. The gap electric field supplies this energy over repeated crossings.
Watch out: The magnetic field bends its path; the electric field across the gap transfers energy.