Calculator Inputs
Formula Used
Perpendicular speed: v⊥ = v sin(θ)
Larmor radius: rL = γ m v⊥ / (|q| B)
Cyclotron angular frequency: ωc = |q| B / (γ m)
Non-relativistic energy relation: v = √(2K / m)
Relativistic energy relation: γ = 1 + K / (m c²), then v = c √(1 - 1 / γ²)
The Larmor radius gives the curvature scale of a charged particle moving across a magnetic field. Only the velocity component perpendicular to the field bends into circular motion. The parallel component remains unchanged and creates a helical path.
The charge sign changes the direction of rotation, but not the radius magnitude. Stronger fields, larger charges, or smaller perpendicular speeds reduce the radius.
How to Use This Calculator
- Select a particle preset or choose a custom particle.
- Enter mass, charge, magnetic field, and the pitch angle relative to the field.
- Choose whether to calculate from speed or kinetic energy.
- Enable relativistic mode for high-energy particles or speeds near light speed.
- Press Calculate to show the result summary above the form, then export the values if needed.
Example Data Table
These examples show how different particles, energies, and fields change the gyroradius.
| Case | Particle | Mode | Field | Input | Angle | Radius |
|---|---|---|---|---|---|---|
| Electron beam | Electron | Velocity / Classical | 0.2 T | v = 1.5×10^7 m/s | 90° | 426.422258 µm |
| Proton in lab field | Proton | Energy / Classical | 0.5 T | K = 20 keV | 60° | 3.539438 cm |
| Alpha particle | Alpha particle | Velocity / Classical | 1.2 T | v = 3×10^6 m/s | 45° | 3.665702 cm |
| Relativistic electron | Electron | Energy / Relativistic | 2 T | K = 500 keV | 80° | 1.432824 mm |
Frequently Asked Questions
1) What is the Larmor radius?
It is the radius of the circular part of a charged particle’s motion inside a magnetic field. It depends on perpendicular speed, particle momentum, charge magnitude, and field strength.
2) Why does pitch angle matter?
Pitch angle determines how much of the total speed is perpendicular to the magnetic field. A larger perpendicular component makes a larger circular orbit and changes the helix shape.
3) Why do electrons and protons give different radii?
Even with the same speed and field, different particle masses change momentum. Because the radius depends on momentum divided by charge magnitude, heavy particles usually curve less tightly.
4) When should I use relativistic mode?
Use it when particle speed is a significant fraction of light speed or when kinetic energy is high enough that classical formulas become inaccurate. It adjusts momentum through the Lorentz factor.
5) What happens if the magnetic field doubles?
If all other inputs stay fixed, the Larmor radius is cut in half. Stronger magnetic fields bend the particle more sharply and reduce the orbit size.
6) Why did my result become zero?
A zero radius usually means the perpendicular velocity is zero. That happens when the pitch angle is 0° or 180°, so the particle moves along the field instead of orbiting around it.
7) Does charge sign affect the radius value?
The sign changes rotation direction, but the radius magnitude uses the absolute charge value. Positive and negative particles with the same mass, speed, and |q| have equal radius magnitudes.
8) Can I calculate from energy instead of speed?
Yes. Choose the energy-based method, enter kinetic energy, and the calculator converts that input into particle speed before finding perpendicular motion, gyrofrequency, and radius.