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Formulas/physics/Moving Charges/Cyclotron Frequency

Cyclotron Frequency

Natural frequency of circular orbit in B. The resonance condition in a cyclotron requires the alternating field frequency to match ν_c. Speed-independent up to relativistic effects.
Class 12
Derivation

From Period to Frequency

From mc04, T=2πm/qBT = 2\pi m / qB. The cyclotron frequency:

νc=1T=qB2πm\boxed{\nu_c = \frac{1}{T} = \frac{qB}{2\pi m}}

Angular form:

ωc=2πνc=qBm\boxed{\omega_c = 2\pi\nu_c = \frac{qB}{m}}

Speed Independence

Neither νc\nu_c nor ωc\omega_c depends on the particle's speed. This is the foundation of cyclotron operation.

Cyclotron Resonance Condition

For a cyclotron to work, the alternating electric field between the dees must oscillate at exactly νc\nu_c. The particle arrives at the gap every half-period T/2T/2, receiving a kick each time.

Numerical Example

For a proton (q=1.6×10−19q = 1.6 \times 10^{-19} C, m=1.67×10−27m = 1.67 \times 10^{-27} kg) in B=1B = 1 T:

νc=1.6×10−19×12π×1.67×10−27≈15.2 MHz\nu_c = \frac{1.6 \times 10^{-19} \times 1}{2\pi \times 1.67 \times 10^{-27}} \approx 15.2\,\text{MHz}
Warning
At relativistic speeds, the rest mass must be replaced by relativistic mass $\gamma m_0$, making $\nu_c$ speed-dependent. Classical cyclotrons are limited to non-relativistic particles for this reason.