Cyclotron Frequency
Calculate the cyclotron frequency f_c = qB/(2πm) of charged particles in a magnetic field. Free online physics calculator with charts, breakdowns, and support for electrons, protons, and more.
About This Calculator
About Cyclotron Frequency
The Cyclotron Frequency Calculator computes the orbital frequency of a charged particle moving perpendicular to a uniform magnetic field. This frequency, also known as the gyrofrequency, is fundamental to the operation of cyclotron particle accelerators and appears in plasma physics, astrophysics, and nuclear magnetic resonance (NMR).
The cyclotron frequency is derived by balancing the Lorentz magnetic force F = qvB with the centripetal force F = mv²/r, giving the angular frequency ω = qB/m. Dividing by 2π gives f_c = qB/(2πm). Remarkably, the frequency is independent of the particle's velocity and orbital radius — a property that makes cyclotrons efficient accelerators.
This calculator supports electrons, protons, deuterons, alpha particles, and muons with their standard charge and mass values built in. For other particles, select the custom option and enter the charge and mass manually. You can optionally specify the orbital radius to compute the tangential velocity of the particle.
Applications Across Disciplines
Nuclear Physics: Cyclotrons accelerate protons and ions for collision experiments, neutron generation, and studying nuclear structure. Medical cyclotrons produce radioisotopes for PET imaging and cancer therapy.
Plasma Physics: The cyclotron frequency determines how charged particles gyrate around magnetic field lines in fusion reactors, space plasmas, and the Earth's magnetosphere, influencing plasma confinement and wave-particle interactions.
Astrophysics: Cyclotron and synchrotron radiation from electrons spiraling in magnetic fields provides key observational signatures from pulsars, supernova remnants, and active galactic nuclei across the electromagnetic spectrum.
Medical Imaging: The related Larmor frequency forms the basis of magnetic resonance imaging (MRI), where hydrogen proton spins precess at frequencies proportional to the applied magnetic field strength.
Formula
f_c = qB / (2πm)
Where f_c is the cyclotron frequency (Hz), q is the particle charge (C), B is the magnetic flux density (T), and m is the particle mass (kg). The angular frequency is ω_c = qB/m = 2πf_c, and the orbital period is T = 1/f_c.
Frequently Asked Questions
What is cyclotron frequency?
Cyclotron frequency is the frequency at which a charged particle revolves in a uniform magnetic field perpendicular to its direction of motion. It is given by the formula f_c = qB/(2πm), where q is the particle's charge, B is the magnetic field strength, and m is the particle's mass.
How do you calculate cyclotron frequency?
Cyclotron frequency is calculated using the formula f_c = qB/(2πm). First determine the particle's charge q in coulombs and mass m in kilograms, then multiply by the magnetic field strength B in teslas. Divide the product by 2πm to get the frequency in hertz.
What is the cyclotron frequency of an electron in a 1 T field?
The cyclotron frequency of an electron in a 1 T magnetic field is approximately 28 GHz (2.8 × 10¹⁰ Hz). Using the formula f_c = qB/(2πm) with electron charge 1.602 × 10⁻¹⁹ C and mass 9.109 × 10⁻³¹ kg gives f_c ≈ 2.8 × 10¹⁰ Hz.
What is the cyclotron frequency of a proton in a 1 T field?
The cyclotron frequency of a proton in a 1 T magnetic field is approximately 15.4 MHz (1.54 × 10⁷ Hz). Using the formula f_c = qB/(2πm) with proton charge 1.602 × 10⁻¹⁹ C and mass 1.673 × 10⁻²⁷ kg.
Does cyclotron frequency depend on velocity or radius?
No, cyclotron frequency depends only on the particle's charge-to-mass ratio and the magnetic field strength. It is independent of the particle's velocity and the radius of its orbit, which is why cyclotrons can accelerate particles without retuning the RF frequency.
What is a cyclotron used for?
Cyclotrons are particle accelerators used in nuclear physics research, medical isotope production for PET scans and cancer treatment, materials science, and particle therapy. They accelerate charged particles using a constant magnetic field and a rapidly alternating electric field.
What is the difference between cyclotron frequency and Larmor frequency?
Cyclotron frequency f_c = qB/(2πm) describes the orbital motion of a free charged particle in a magnetic field. Larmor frequency (or precession frequency) describes the spin precession of a magnetic dipole in a magnetic field. In NMR and MRI, the Larmor frequency depends on the gyromagnetic ratio of the nucleus.
What happens to cyclotron frequency at relativistic speeds?
At relativistic speeds, the particle's effective mass increases according to m = m₀/√(1-v²/c²), causing the cyclotron frequency to decrease. This is the fundamental limitation of classical cyclotrons and why synchrotrons or synchrocyclotrons are needed for high-energy particles.