What is the Larmor radius formula?

The Larmor (cyclotron or gyro) radius is r = mv / (|q|B), where m is the particle mass in kg, v is the component of velocity perpendicular to the field in m/s, q is the particle charge in coulombs, and B is the magnetic field strength in tesla. A larger speed or smaller charge or weaker field all increase the radius.

What is the difference between Larmor radius, cyclotron radius, and gyroradius?

They are the same quantity with different names depending on the field. Larmor radius is common in plasma physics, cyclotron radius in accelerator physics, and gyroradius in space and astrophysics.

Does this formula work at relativistic speeds?

This calculator uses the classical formula r = mv/(|q|B). At speeds approaching c, the relativistic momentum p = γmv should replace mv, giving r = γmv/(|q|B). The classical result is a good approximation for v below about 10% of c.

What happens if the velocity is not perpendicular to the field?

Only the velocity component perpendicular to the field (v⊥) contributes to circular motion. The parallel component causes the particle to spiral along field lines, tracing a helix. The Larmor radius is computed using v⊥ only.

What is the cyclotron frequency?

The cyclotron frequency (angular frequency) is ω = |q|B/m, and the ordinary frequency is f = |q|B/(2πm). Notably, neither depends on the particle's speed — faster particles travel larger circles in the same period, which is the key principle behind the cyclotron accelerator.

Larmor Radius Calculator

Name: Larmor Radius Calculator
Availability: InStock
Author: Nham Vu

Enter particle mass, charge, velocity, and magnetic field strength to get the Larmor radius r = mv / (|q|B).

Input Parameters

Particle Preset

Speed (v_⊥)

Magnetic Field (B)

Results

Enter values and click Calculate

Summary

Enter particle mass, charge, velocity, and magnetic field strength to get the Larmor radius r = mv / (|q|B).

How it works

Select a particle preset or enter custom mass and charge values.
Enter the particle speed and choose the unit (m/s, km/s, or % of c).
Enter the magnetic field strength and select the unit (T, mT, or µT).
The Larmor radius r = mv / (|q|B) is calculated instantly.
The period and cyclotron frequency are also shown in the results.

Use cases

Determine the orbit radius of electrons in a cyclotron or synchrotron.
Estimate the gyroradius of solar wind protons in Earth's magnetosphere.
Solve introductory plasma physics and particle-accelerator problems.
Verify textbook answers for charged-particle motion in uniform magnetic fields.
Calculate alpha particle trajectories in nuclear physics experiments.
Explore how increasing the magnetic field reduces the orbital radius.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu