Cyclotron Frequency Calculator

A charged particle in a magnetic field moves in a circle with a frequency that depends only on charge-to-mass ratio and field strength — not on speed. This is the basis of cyclotrons.

⚛️ Particle Physics📐 f = qB/2πm🧲 Magnetism

Charge q (C)

Mass m (kg)

Magnetic Field B (T)

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Cyclotron Frequency	f_c	f_c = qB/(2πm)	Hz
Orbital Radius	r	r = mv/(qB)	m
Period	T	T = 2πm/(qB)	s
Angular Frequency	ω_c	ω_c = qB/m	rad/s

Step-by-Step Examples

Example 1

Proton in 1 T field

Proton: q=1.6e-19, m=1.67e-27 kg, B=1 T.

f = qB/(2*pi*m) = 1.6e-19*1/(2*pi*1.67e-27)
f = 1.528e7 Hz = 15.28 MHz

✓ 15.28 MHz proton cyclotron frequency

Example 2

Electron in 1 T field

Electron: q=1.6e-19, m=9.11e-31 kg, B=1 T.

f = 1.6e-19*1/(2*pi*9.11e-31) = 27.99 GHz

✓ 28 GHz — electron cyclotron resonance in microwave range

Example 3

MRI Larmor Frequency

Proton in B=3 T (MRI scanner).

f = 15.28 * 3 = 45.84 MHz

✓ 127 MHz at 3 T MRI field — in RF range

Real-World Applications

🔬

Cyclotron Accelerators

Particles are accelerated at the cyclotron resonance frequency, gaining energy each half-orbit.

🏺

MRI Scanners

Proton Larmor frequency = cyclotron frequency at scanner field. RF pulses at this frequency flip spins.

🧭

Mass Spectrometers

ICR mass spec: different masses have different cyclotron frequencies, enabling ultra-precise mass measurement.

🌌

Space Plasma Physics

Electron cyclotron radiation from Earth radiation belts detected as radio waves.

Common Mistakes to Avoid

⚠️

Frequency is independent of velocity

This is the key property: f_c = qB/(2*pi*m) does not depend on particle speed, only on q/m and B.

⚠️

Relativistic correction needed at high energies

At high speeds, relativistic mass increases, lowering the cyclotron frequency. This is why synchrotrons vary the frequency.

⚠️

Radius does depend on velocity

r = mv/(qB). Radius increases as the particle speeds up, creating the spiral outward in a cyclotron.

Frequently Asked Questions

Why is cyclotron frequency independent of speed? ▾

Higher speed gives larger radius but covers it in the same time. Period T = 2*pi*m/(qB). Larger circle, faster particle, same time.

What is a synchrotron? ▾

A synchrotron compensates for relativistic mass increase by gradually increasing B field as particles accelerate, maintaining the orbital frequency.

How does MRI use cyclotron frequency? ▾

Proton Larmor (cyclotron) frequency at field B: f = 42.58 MHz/T. At 3 T MRI: f = 127.7 MHz. RF pulses at this frequency resonate with proton spins.

What is electron cyclotron resonance? ▾

Electrons in magnetic field resonate at ~28 GHz/T. Used in ECR ion sources and microwave amplifiers.

What is a mass spectrometer using cyclotron? ▾

ICR (Ion Cyclotron Resonance) MS: ions in magnetic field orbit at f = qB/(2*pi*m). Frequency directly gives mass-to-charge ratio with sub-ppm accuracy.

Related Physics Calculators

Formula Explorer connections

Interpretation: This relationship connects frequency, wavelength, speed, phase, intensity or resonance in an oscillating system. Assumption: Identify the medium, boundary conditions and reference frame. Linear waves, small amplitudes, nondispersive media or ideal resonance may be assumed.

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