Larmor Radius (Gyroradius) Calculator
Find the Larmor radius (gyroradius) of a charged particle orbiting a magnetic field line, r_L = mv⊥/(qB).
🌀 What is the Larmor Radius (Gyroradius) Calculator?
This Larmor radius calculator finds the radius of the circular orbit a charged particle traces as it gyrates around a magnetic field line. Enter the particle's perpendicular velocity, the magnetic field strength, and choose a particle, and it returns the orbit radius, the gyration angular frequency, and the orbit period.
A charged particle moving across a magnetic field feels a continuous sideways (Lorentz) force that curves its path into a circle rather than a straight line. The radius of that circle, r_L = mv⊥/(qB), depends on the particle's mass and perpendicular speed (which widen the orbit) and its charge and the field strength (which shrink it).
This single length scale is central to magnetic confinement fusion (where the plasma must be much larger than the ion Larmor radius for confinement to work) and to space physics (where the vastly different Larmor radii of electrons and ions, and their scaling with the local magnetic field, shape everything from the solar wind to Earth's radiation belts).
This calculator is useful for plasma physics and fusion engineering students, and anyone studying charged particle motion in magnetic fields, from lab experiments to space plasmas.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Electron in a 1 tesla lab field
Example 2 - Proton in a 5 tesla tokamak field
Example 3 - Electron in the solar wind
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Larmor radius?
The Larmor radius (also called gyroradius) is the radius of the circular orbit a charged particle traces out as it gyrates around a magnetic field line, due to the magnetic (Lorentz) force continuously bending its perpendicular velocity into a circle. It is one of the most basic length scales in plasma and space physics.
What is the formula for Larmor radius?
r_L = m v⊥ / (q B), where m is the particle's mass, v⊥ is its velocity component perpendicular to the magnetic field, q is its charge, and B is the magnetic field strength. Heavier or faster particles have larger orbits; stronger fields or larger charges shrink the orbit.
Why does only the perpendicular velocity matter?
The magnetic force on a moving charge, q v × B, is always perpendicular to both the velocity and the field, so it can only curve motion in the plane perpendicular to B. Any velocity component parallel to the field is unaffected by the magnetic force and simply carries the particle in a straight line along the field, turning the overall motion into a helix rather than a pure circle.
Why do electrons have a smaller Larmor radius than protons?
For the same speed and magnetic field, the Larmor radius is directly proportional to mass, and a proton is about 1836 times more massive than an electron. This means protons gyrate on orbits about 1836 times larger (and correspondingly slower) than electrons under the same conditions.
How is Larmor radius related to cyclotron frequency?
They describe the same circular motion from two angles: cyclotron frequency ω_c = qB/m is how fast the particle completes its orbit, while Larmor radius r_L = v⊥/ω_c is how big that orbit is. The related <a href="/science/plasma-physics/cyclotron-frequency-calculator/">Cyclotron Frequency Calculator</a> computes the frequency directly.
Why does Larmor radius matter for fusion confinement?
Magnetic confinement fusion relies on strong magnetic fields to keep charged particles gyrating in small orbits far smaller than the device itself, preventing them from simply drifting to the walls. If the Larmor radius were comparable to the plasma's size, the magnetic confinement would fail to hold the particles in.
How large is the Larmor radius in different settings?
An electron in a 1 tesla laboratory field with a typical thermal speed gyrates on a radius of a few micrometres, while a solar wind electron in the much weaker interplanetary magnetic field (a few nanotesla) can have a Larmor radius of hundreds of metres. The much larger space-physics radius reflects just how weak interplanetary magnetic fields are compared to laboratory magnets.
Does Larmor radius depend on the direction of motion along the field?
No, only the perpendicular velocity component sets the orbit radius. A particle's parallel velocity (along the field line) determines how far it travels along the helix per orbit, but has no effect on how wide the circular part of that helix is.
What happens to Larmor radius as the magnetic field gets stronger?
Larmor radius shrinks in direct proportion to 1/B, so doubling the magnetic field strength halves the orbit radius. This is one reason fusion reactors push toward the strongest practical magnetic fields, to shrink particle orbits and improve confinement within a given device size.
Is the Larmor radius the same as the plasma's overall size?
No, they are very different scales. Larmor radius is a microscopic length describing a single particle's orbit around a field line, while the plasma's overall size (its macroscopic dimension) needs to be much larger than the Larmor radius for magnetic confinement to work at all.