Larmor Radius Calculator

What the Larmor radius means in a magnetic field

For a charged particle in a uniform magnetic field, the Larmor radius is the radius of the circle traced by the part of the motion that is perpendicular to the field. If the particle also moves along the field, the full trajectory becomes a helix, but the gyroradius still describes the cross-field circle wrapped around the field line.

That radius is a quick measure of how strongly the particle is magnetized. When the gyroradius is much smaller than the size of the device, plasma region, or beamline feature you care about, the particle tends to stay tied to the field structure. When the radius is large compared with the system scale, the motion is less confined and the particle can sample a much wider area.

This calculator keeps the relationship practical by letting you solve for any one of the five linked quantities—mass, charge magnitude, perpendicular velocity, magnetic field strength, or Larmor radius—so long as the other four are entered in SI units.

Larmor radius formula and rearrangements

For non-relativistic motion strictly perpendicular to a uniform magnetic field, the Larmor radius r is

Algebraic form:

r = (m · v) / (|q| · B)

where

• r is the Larmor radius in meters (m)
• m is the particle mass in kilograms (kg)
• v is the component of velocity perpendicular to the magnetic field in meters per second (m/s)
• q is the particle charge in coulombs (C), and the formula uses the magnitude |q|
• B is the magnetic field strength in tesla (T)

In SI units, this expression comes from balancing the magnetic part of the Lorentz force against the centripetal force needed for circular motion. The magnetic force magnitude is |q|vB, while the circular-motion requirement is mv²/r. Equating them and canceling one factor of v gives the usual gyroradius formula.

The same relationship can be written in MathML as:

Because the equation is linear in mass and perpendicular speed and inverse in charge magnitude and field strength, it can be rearranged in several useful ways. If the radius is known, the field can be found from B = (m · v) / (|q| · r). If the field and radius are known, the perpendicular speed follows from v = (r · |q| · B) / m. The calculator performs those rearrangements automatically once you leave exactly one input blank.

How to use the Larmor radius calculator

This Larmor radius calculator expects SI units, so the easiest workflow is to decide which quantity you want to solve and enter the other four. Fill in the known values, leave the unknown field blank, and submit the form.

Use kilograms for mass, coulombs for charge magnitude, meters per second for the perpendicular speed, tesla for magnetic field strength, and meters for the radius. If your starting values are in gauss, electronvolts, centimeters, or any other unit system, convert them before entering them because this page does not perform unit conversion.

1. Choose the quantity to solve for: m, |q|, v, B, or r.
2. Enter the four known values in SI units.
3. Leave exactly one field blank.
4. Submit the form to calculate the missing quantity.

If you leave more than one field empty, or if you fill every field, the calculator will prompt you to leave exactly one blank. That rule matters because the underlying gyroradius relation gives one unknown from one equation. Results are shown in scientific notation so very small radii and very large radii stay readable.

How to interpret the Larmor radius result

The number returned here is the size of the circular part of the particle’s path in the magnetic field. A small Larmor radius means tight bending and strong magnetic control over the trajectory. A large radius means the orbit is broad and the particle can move across a wider region before completing a turn.

The scaling rules are useful for intuition. The radius increases in direct proportion to mass and perpendicular speed, so doubling either one doubles the radius. The radius decreases when the charge magnitude or magnetic field strength increases, so doubling either one cuts the radius in half. Those proportional trends are often enough to judge whether a change will improve confinement or allow a particle population to spread farther.

• Small radius: tight orbit, stronger magnetic confinement, and motion that tends to follow field lines closely on larger scales.
• Large radius: broad orbit, weaker confinement, and greater ability to sample spatial variations or escape a bounded region.
• Charge sign: changes the direction of rotation, but not the numerical radius returned by this calculator.

Remember that the calculator uses only the velocity component perpendicular to the field. If your particle speed is split between perpendicular and parallel motion, enter only the perpendicular part here; otherwise the radius you calculate will be too large for the circular portion of the path.

Worked example: electron gyroradius in a laboratory field

Consider an electron moving perpendicular to a uniform magnetic field of 0.1 T with a speed of 1.0 × 10⁶ m/s. We want to compute its Larmor radius. This is a useful example because the numbers are realistic for a laboratory estimate and the result is small enough to show how strongly electrons can be bent by even modest magnetic fields.

Known values:

• Electron mass: m ≈ 9.11 × 10^-31 kg
• Electron charge magnitude: |q| ≈ 1.60 × 10^-19 C
• Velocity: v = 1.0 × 10⁶ m/s
• Magnetic field: B = 0.1 T

Step 1: write the radius formula.

r = (m · v) / (|q| · B)

Step 2: multiply mass and velocity. That gives m · v = 9.11 × 10^-25 kg·m/s. Step 3: multiply charge magnitude and magnetic field. That gives |q| · B = 1.60 × 10^-20 C·T. Step 4: divide the two results.

r = (9.11 × 10^-25) / (1.60 × 10^-20) m ≈ 5.69 × 10^-5 m

So the electron’s Larmor radius is about 5.7 × 10^-5 m, or roughly 57 micrometers. In the calculator, you would enter the mass, charge magnitude, speed, and field strength, leave the radius box blank, and compute the result. If you wanted to solve the inverse problem instead, you could enter that radius, keep the other known values, leave the velocity field blank, and recover the same speed.

Comparison examples: electron and proton Larmor radii

Comparing a few Larmor-radius cases side by side makes the proportionality easier to see. The table below uses representative values to show why electrons usually have much tighter orbits than protons when the speed and magnetic field are similar.

The pattern is what matters: in strong laboratory fields, electrons can have radii measured in micrometers or millimeters, while in weak space-plasma fields the radius can grow dramatically. Because protons are much heavier than electrons but carry the same charge magnitude, they usually curve far less for the same speed and field strength.

Where the Larmor radius matters

The Larmor radius appears any time a magnetic field controls charged-particle motion, because it translates particle properties into a directly usable length scale. That makes it one of the fastest ways to judge confinement, curvature, and cross-field transport.

Magnetically confined fusion

In tokamaks and stellarators, electrons and ions spiral around field lines while collisions and collective plasma effects act on longer scales. Engineers want the gyroradius to stay much smaller than the machine dimensions so particles and energy remain magnetically guided. This calculator is handy for quick checks when you want to see whether a proposed field strength keeps the radius comfortably below the plasma size.

Space and astrophysical plasmas

In Earth’s magnetosphere, the solar wind, planetary bow shocks, and many astrophysical environments, gyroradius estimates help show whether a particle stays tied to a field structure or crosses it more freely. A radius that is tiny compared with the system scale often justifies guiding-center reasoning, while a comparable radius means the particle is sampling a much larger region.

Charged-particle beams and detectors

In accelerators, mass spectrometers, and tracking detectors, curvature in a magnetic field is tied directly to momentum and charge. The same physics behind the Larmor radius underlies magnetic rigidity and the bending of tracks, so the formula is still useful even when a full beamline model eventually needs more detail.

Assumptions and limitations of the Larmor radius formula

This calculator is built for straightforward estimates and educational checks, not for every possible plasma or beam situation. The simple single-particle model works well only when its assumptions are respected.

• Uniform magnetic field: the field is assumed to be uniform across the orbit. Strong spatial variation on scales similar to the radius reduces the accuracy of the simple model.
• Perpendicular motion: the radius is based on the velocity component perpendicular to the field. If the particle also moves along the field, the full path is helical rather than purely circular.
• Non-relativistic speeds: the formula assumes speeds well below the speed of light. At relativistic energies, momentum is not simply m v, and a relativistic treatment should be used.
• Charge magnitude: the calculator uses |q| and therefore returns the magnitude of the radius only. The sign still matters for the direction of gyration.
• SI units only: all values are expected in kg, C, m/s, T, and m.

For precision design work in complex geometries—such as full plasma simulations, detailed beam-dynamics studies, or highly relativistic astrophysical scenarios—you will need more advanced tools. Still, the formula here remains one of the most useful first checks in charged-particle physics because it turns several physical ideas into one interpretable length scale.