Larmor Radius Calculator

What is the Larmor radius (gyroradius)?

The Larmor radius, also called the gyroradius, is the radius of the circular path traced by a charged particle moving perpendicularly to a uniform magnetic field. It is a fundamental length scale in plasma physics, space physics, and charged-particle beam design, because it tells you how tightly a magnetic field can bend a particle’s trajectory.

When an electron, proton, or ion enters a magnetic field with some velocity, it experiences the Lorentz force. If the velocity is perpendicular to the magnetic field, this force acts as a centripetal force and makes the particle follow a circular orbit around the magnetic field line. The stronger the magnetic field or the larger the charge, the tighter (smaller) the orbit. The larger the mass or speed, the larger the orbit.

This calculator lets you work with the standard non-relativistic Larmor radius formula. You can solve for any one of the five quantities in the relationship—mass, charge magnitude, velocity, magnetic field, or orbit radius—by entering the other four.

Larmor radius formula

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 (m)
• m is the particle mass (kg)
• v is the component of velocity perpendicular to the magnetic field (m/s)
• q is the particle charge (C); the formula uses the magnitude |q|
• B is the magnetic field strength (tesla, T)

In SI units, this comes directly from equating the magnetic part of the Lorentz force to the centripetal force required for circular motion. The magnetic force magnitude for perpendicular motion is |q| v B, and the centripetal force needed for a mass m moving in a circle of radius r at speed v is m v² / r. Setting these equal and solving for r gives the formula above.

The same relationship can be written in MathML as:

The calculator rearranges this same expression to solve for whichever one of the five quantities you leave blank.

How to use the Larmor radius calculator

The calculator assumes SI units throughout. To perform a calculation:

1. Decide which quantity you want to compute: m, |q|, v, B, or r.
2. Enter known values for the other four fields, using SI units:

• Mass m in kilograms (kg)
• Charge magnitude |q| in coulombs (C)
• Velocity v in meters per second (m/s)
• Magnetic field B in tesla (T)
• Radius r in meters (m)

3. Leave exactly one input field blank—the one you want the calculator to solve for.
4. Click the compute button.

If more than one field is empty, or if all are filled, the calculator will report an error and ask you to provide exactly four known values. The result is given as a single number in SI units.

Interpreting the results

The magnitude of the Larmor radius tells you how tightly a magnetic field confines or bends a particle’s motion.

• Small radius: the particle’s orbit is tight. The magnetic field dominates its motion, and guiding-center approximations (following the field lines) are usually valid on larger scales.
• Large radius: the particle moves in a wide arc, sampling a larger region of space. Magnetic confinement becomes weaker, and the particle may escape a device or region.

Because the formula is linear in both mass and velocity, and inversely proportional to both |q| and B, you can qualitatively predict how the radius will change if you adjust one parameter:

• Doubling the mass m doubles the radius r.
• Doubling the perpendicular speed v also doubles r.
• Doubling the charge magnitude |q| halves r.
• Doubling the magnetic field B halves r.

Note that the formula uses the magnitude of the charge, |q|. The sign of the charge (positive or negative) determines the direction of rotation around the magnetic field line (clockwise vs. counter-clockwise) but does not change the radius. The calculator therefore always returns a positive value for the radius.

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.

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 (perpendicular component)
• Magnetic field: B = 0.1 T

Step 1: Write the formula for the radius:

r = (m · v) / (|q| · B)

Step 2: Multiply mass and velocity:

m · v = (9.11 × 10^-31 kg) × (1.0 × 10⁶ m/s) = 9.11 × 10^-25 kg·m/s

Step 3: Multiply charge magnitude and magnetic field:

|q| · B = (1.60 × 10^-19 C) × (0.1 T) = 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 Larmor radius is about 5.7 × 10^-5 m, or 57 micrometers. In the calculator, you would enter m, |q|, v, and B with these values, leave the radius field blank, and let the tool confirm this result.

You can then perform the inverse problem. If you know that an electron orbit in a 0.1 T field has a radius of 5.7 × 10^-5 m and you know m and |q|, you can leave the velocity field blank and solve for v.

Comparison examples for different particles and fields

The table below compares typical Larmor radii for electrons and protons in a range of representative environments. All entries assume perpendicular motion and non-relativistic speeds.

These examples highlight several useful trends:

• In strong laboratory fields (0.1 T and above), low-energy electrons can have gyroradii on the order of micrometers to millimeters.
• In very weak interplanetary or interstellar magnetic fields (nanotesla to microtesla), high-energy particles can have gyroradii of kilometers to hundreds of thousands of kilometers.
• Heavier particles such as protons have much larger radii than electrons at the same speed and field because of their larger mass.

Applications of the Larmor radius

The Larmor radius appears in many areas of physics and engineering.

Magnetically confined fusion

In tokamaks and stellarators, charged particles in the plasma spiral around magnetic field lines. Engineers aim to keep the Larmor radius much smaller than the device size so that particles remain well confined. The calculator can help estimate the field strength or allowable particle energies needed to achieve a given confinement scale.

Space and astrophysical plasmas

In Earth’s magnetosphere and the solar wind, the gyroradius is used to decide whether particles are tied to magnetic field lines or can move freely across them. For example, a 10 keV electron in the Earth’s magnetotail may have a gyroradius of only tens of kilometers in a tens-of-nanotesla field, far smaller than typical magnetospheric dimensions. This justifies modeling their motion as being closely linked to the field.

Charged-particle beams and detectors

In accelerators and beamlines, magnets bend charged-particle trajectories according to their momentum and charge. The Larmor radius is closely related to the concept of magnetic rigidity. Detector systems such as bubble chambers or tracking detectors infer charge-to-mass ratios from the curvature of particle tracks in known magnetic fields, which is directly tied to the gyroradius formula.

Assumptions and limitations

This calculator is designed for clear, educational estimates and simple engineering calculations. It relies on several important assumptions:

• Uniform magnetic field: The field is taken to be spatially uniform over the particle’s orbit. Real fields can vary in space and time; if the scale of variation is comparable to or smaller than the Larmor radius, this simple model becomes inaccurate.
• Perpendicular motion: The formula uses the velocity component perpendicular to the magnetic field. If the particle has a pitch angle (a component of velocity along the field), the motion is helical; the computed radius still applies to the circular component, but the total path is a helix.
• Non-relativistic speeds: The expression assumes that the particle’s speed is much less than the speed of light, so that relativistic effects on mass and momentum can be ignored. At very high energies, a relativistic generalization using relativistic momentum p = γ m v should be used instead.
• Magnitude of charge: The calculation uses |q|, so it returns the magnitude of the radius only. The sign of the charge affects the sense of rotation but not the numerical radius.
• SI units only: All inputs and outputs are in SI units (kg, C, m/s, T, m). If your data are in electronvolts (eV) or gauss (G), you must first convert them to SI before using the tool.

For precision design work in complex geometries—such as full plasma simulations, detailed beam-dynamics studies, or highly relativistic astrophysical scenarios—this simple single-particle, uniform-field model is not sufficient. In those cases, specialized simulation codes or more advanced analytical treatments are required. The calculator here is best viewed as a convenient way to build intuition and obtain quick order-of-magnitude estimates.