Cyclotron Frequency Calculator
Find the cyclotron frequency of a charged particle gyrating around a magnetic field line using ω_c = qB/m.
🧲 What is the Cyclotron Frequency Calculator?
This cyclotron frequency calculator finds how fast a charged particle gyrates around a magnetic field line. Enter the magnetic field strength and choose a particle, and it returns the cyclotron angular frequency, the ordinary frequency in hertz, and the orbit period.
A charged particle moving across a magnetic field is continuously pushed sideways by the Lorentz force, tracing a circular orbit. The rate of that orbit, ω_c = qB/m, depends only on the particle's charge-to-mass ratio and the field strength, remarkably, it does not depend on the particle's speed at all.
This frequency has direct practical importance: the electron cyclotron frequency (exactly 28 GHz per tesla) sets the microwave frequency used for electron cyclotron resonance heating in fusion plasmas, and the same physics underlies the cyclotron particle accelerator and space-physics phenomena like whistler waves in the ionosphere.
This calculator is useful for plasma physics and fusion engineering students, and anyone studying particle accelerators, ECR heating, or charged particle motion in magnetic fields.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Electron cyclotron resonance (ECR) at 1 tesla
Example 2 - Proton in a 5 tesla tokamak field
Example 3 - Electron in Earth's magnetic field
❓ Frequently Asked Questions
🔗 Related Calculators
What is cyclotron frequency?
Cyclotron frequency (or gyrofrequency) is the rate at which a charged particle orbits around a magnetic field line, due to the magnetic (Lorentz) force continuously curving its perpendicular motion into a circle. It is named after the cyclotron particle accelerator, which uses this exact resonance to accelerate charged particles.
What is the formula for cyclotron frequency?
ω_c = qB/m (angular frequency) or f_c = qB/(2πm) (ordinary frequency), where q is the particle's charge, B is the magnetic field strength, and m is its mass. Unlike many other frequencies in physics, it does not depend on the particle's speed at all.
Why doesn't cyclotron frequency depend on speed?
A faster particle traces a proportionally larger circle (a bigger Larmor radius) but takes exactly the same time to complete one orbit, because both the magnetic force and the required centripetal force scale the same way with speed. The two effects cancel exactly, leaving the orbital frequency independent of velocity.
Why is 28 GHz per tesla such an important number?
28 GHz per tesla is the electron cyclotron frequency, and it sets the exact microwave frequency needed for electron cyclotron resonance (ECR) heating, a standard technique for heating fusion plasmas by driving microwaves at the frequency electrons naturally gyrate at in the local magnetic field. Fusion device magnets are often specified partly around matching this resonance to available microwave sources.
How much lower is the ion cyclotron frequency than the electron cyclotron frequency?
Since cyclotron frequency scales as 1/m, and a proton is about 1836 times more massive than an electron, the proton cyclotron frequency is about 1836 times lower than the electron cyclotron frequency at the same magnetic field. This large separation is exploited in ion cyclotron resonance heating, which uses much lower frequency waves than electron heating.
What is electron cyclotron resonance (ECR)?
Electron cyclotron resonance occurs when an applied electromagnetic wave's frequency matches the local electron cyclotron frequency, allowing efficient energy transfer from the wave to the gyrating electrons. It is used both for plasma heating in fusion research and, at lower power, for generating and sustaining plasmas in ECR ion sources and plasma processing equipment.
How is cyclotron frequency related to the Larmor radius?
They describe the same circular orbit from two angles: cyclotron frequency is how fast the particle goes around, and Larmor radius is how big the orbit is. The related <a href="/science/plasma-physics/larmor-radius-gyroradius-calculator/">Larmor Radius Calculator</a> computes the orbit size for the same particle and field.
How does Earth's magnetic field affect radio waves?
Earth's magnetic field at ionospheric altitudes (tens of microtesla) gives electrons a cyclotron frequency of roughly 1 to 1.5 megahertz, right in the medium-wave radio band. This resonance shapes whistler-mode wave propagation in the ionosphere and magnetosphere, a well-studied topic in space physics.
Does cyclotron frequency depend on the particle's kinetic energy?
No, only on its charge-to-mass ratio and the local magnetic field strength. Two electrons with very different speeds in the same magnetic field gyrate at exactly the same cyclotron frequency, just tracing circles of different sizes (different Larmor radii).
What is the practical use of a cyclotron particle accelerator?
A cyclotron accelerates charged particles by applying an oscillating electric field synchronized to their cyclotron frequency, giving them a small energy kick each time they cross a gap, while a magnetic field keeps them on an outward spiral. Because cyclotron frequency does not change with speed, a single fixed driving frequency can accelerate particles across a wide range of energies, until relativistic mass increase eventually breaks the synchronization at very high energy.