Larmor Radius Calculator
Introduction
This calculator estimates the Larmor radius, also called the gyroradius, for a charged particle moving through a magnetic field. In plain language, it tells you how wide the particle’s circular orbit becomes when the magnetic field bends its motion. That one length scale shows up everywhere from classroom plasma examples to fusion devices, auroras, cosmic-ray transport, and beamline magnets. If you know four of the five quantities in the standard relationship, this tool solves the fifth one for you.
The most important physical idea is simple: a magnetic field does not usually slow a charged particle down, but it can redirect the particle. When the particle’s velocity is perpendicular to the field, the magnetic force points sideways and continuously turns the motion into a circle. A stronger field bends the path more tightly, which means a smaller radius. A heavier particle or a faster perpendicular speed resists that bending more, which means a larger radius. The sign of the charge matters for the direction of rotation, but the size of the orbit depends on the magnitude of the charge.
This page focuses on the standard non-relativistic formula and assumes the particle’s speed component of interest is the component perpendicular to the magnetic field. That makes the calculator especially useful for quick checks, order-of-magnitude estimates, and intuition building. If you are working backwards from an observed orbit radius, you can also use the same relationship to solve for magnetic field strength, speed, mass, or charge magnitude instead of radius.
What the Larmor radius means
The Larmor radius is the radius of the circular path traced by a charged particle moving perpendicularly to a uniform magnetic field. In many practical situations the particle does not move purely in a circle, because it may also have a velocity component along the magnetic field. In that more general case, the path becomes a helix, and the Larmor radius still describes the size of the circular part wrapped around the field line.
That is why the quantity matters so much in plasma physics and space physics. If the gyroradius is tiny compared with the scale of the device or region you are studying, then the particle is tightly tied to the field lines and magnetic confinement is effective. If the gyroradius becomes large, the particle samples a much wider region of space and can cross field structures more easily. A single number therefore gives quick insight into how magnetized the particle really is.
This calculator lets you work with that idea directly. Instead of only computing the radius, it can solve for any one of the five connected variables—mass, charge magnitude, perpendicular velocity, magnetic field, or radius—so long as you provide the other four in SI units.
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 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 comes from equating the magnetic part of the Lorentz force to the centripetal force required for circular motion. For perpendicular motion, the magnetic force magnitude is |q|vB. The centripetal force needed to keep a mass m moving in a circle of radius r at speed v is mv²/r. Set those expressions equal, cancel one factor of v, and you obtain the familiar gyroradius formula.
The same relationship can be written in MathML as:
Because it is a simple proportional relationship, the same expression can be rearranged in several useful ways. For example, if the radius is known, then the magnetic 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 these rearrangements automatically depending on which field you leave empty.
How to use the calculator
The calculator assumes SI units throughout. A smooth way to use it is to decide first which quantity you want to find, then treat the remaining four as known inputs. Enter those four values, leave the unknown field blank, and click the compute button.
The required units are straightforward: mass in kilograms, charge magnitude in coulombs, perpendicular velocity in meters per second, magnetic field strength in tesla, and radius in meters. If your original data are in other units—such as electronvolts for energy, gauss for field, or centimeters for orbit size—convert them to SI before using the tool. The page does not do those conversions automatically.
- Choose the quantity you want to compute: m, |q|, v, B, or r.
- Enter known values for the other four fields using SI units.
- Leave exactly one field blank.
- Submit the form to calculate the missing quantity.
If more than one field is empty, or if all five fields are filled, the calculator will ask you to leave exactly one field blank. That rule is essential because the formula contains five linked variables but only one independent equation. The result is displayed in scientific notation so that both very small and very large values remain readable.
How to interpret the result
The magnitude of the Larmor radius tells you how tightly a magnetic field confines or bends a particle’s motion. A small radius means the orbit is tight and the field has strong control over the trajectory. A large radius means the particle bends gently and can travel across a wider region before completing an orbit.
The proportional trends are especially helpful when you want intuition without doing a full new calculation. Because the formula is linear in both mass and perpendicular speed, doubling either of those quantities doubles the radius. Because the formula is inversely proportional to charge magnitude and magnetic field strength, doubling either of those cuts the radius in half. These simple scaling rules are often enough to judge whether a design change will improve confinement or whether a particle population will remain magnetized.
- Small radius: tighter orbit, stronger magnetic confinement, and motion that often follows field lines closely on larger scales.
- Large radius: broader 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.
Keep in mind that the formula uses the perpendicular component of velocity. If the total speed is known but only part of it is perpendicular to the field, then only that perpendicular part should be entered here. Otherwise the calculated radius will be too large.
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 good example because the numbers are realistic for a laboratory-scale estimate and the final answer 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 for different particles and fields
Comparing several cases side by side helps show how strongly environment and particle type matter. The table below uses representative values rather than a single exact application, but it clearly shows why electrons are usually much more tightly magnetized than protons at the same speed and field strength.
| Particle | Speed v (m/s) | Magnetic field B (T) | Radius r (m) |
|---|---|---|---|
| Electron | 1 × 106 | 0.1 | 5.7 × 10-5 |
| Electron | 5 × 106 | 1 × 10-5 | ≈ 2.8 m |
| Proton | 1 × 105 | 0.01 | ≈ 0.105 m |
| Proton | 1 × 107 | 5 × 10-9 | ≈ 2.1 × 105 |
Several trends stand out immediately. In strong laboratory fields, low-energy electrons can have radii measured in micrometers to millimeters. In weak interplanetary or astrophysical fields, energetic particles can have gyroradii that grow to kilometers or far more. Heavier particles such as protons typically show much larger radii than electrons because their mass is so much larger, even though the elementary charge magnitude is the same.
Applications of the Larmor radius
The Larmor radius appears in many branches of physics and engineering because it links particle properties directly to magnetic control. It is one of the quickest ways to decide whether a magnetic geometry is likely to confine a population of charged particles or whether those particles are effectively free to drift across the region you care about.
Magnetically confined fusion
In tokamaks and stellarators, ions and electrons spiral around magnetic field lines while collisions and collective effects act on longer scales. Designers want the gyroradius to remain much smaller than the device dimensions, because that helps keep particles and energy from escaping too quickly. The calculator is useful here for sanity checks: if the radius is too large compared with the plasma size, confinement will be poor.
Space and astrophysical plasmas
In Earth’s magnetosphere, the solar wind, planetary bow shocks, and many astrophysical environments, gyroradius estimates help determine whether particles remain attached to field structures or move across them more freely. A particle with a radius tiny compared with the scale of the system can often be treated with guiding-center ideas. A particle with a comparable radius cannot.
Charged-particle beams and detectors
In accelerators, mass spectrometers, and tracking detectors, curvature in a magnetic field is directly tied to momentum and charge. The same basic physics behind the Larmor radius underlies magnetic rigidity and the bending of particle tracks. Even when a practical setup uses more detailed beam optics, the radius formula remains a valuable back-of-the-envelope estimate.
Assumptions and limitations
This calculator is designed for educational estimates and straightforward engineering checks, not for every possible plasma or beam configuration. Several assumptions are built into the simple single-particle formula, and understanding them is the best way to use the result wisely.
- Uniform magnetic field: the field is assumed to be uniform across the orbit. Strong spatial variation on scales comparable 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 total 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.
Enter any four quantities in SI units, leave exactly one field blank, and the calculator will solve for the missing value.
Play the orbit tuner mini-game
Want a fast, visual way to feel the formula instead of only reading it? This optional mini-game turns the same relationship into a short magnetic-confinement challenge. You control the magnetic field strength B. Stronger fields pull the orbit inward, weaker fields let it expand outward, and temporary momentum changes force you to retune on the fly. The goal is to keep the particle’s orbit lined up with the glowing target band, collect blue flux nodes, and avoid red turbulence pockets.
It does not change the calculator’s result at all; it is simply a hands-on intuition builder. Drag or tap across the game area on mobile, or use the left and right arrow keys on desktop. The best runs come from remembering the actual physics: when particle momentum increases, the orbit wants to grow, so you must raise B to hold the same radius. When effective charge influence increases, the orbit contracts and you need less field to stay on target.
Quick rule of thumb during play: if the orbit is too wide, increase B; if the orbit is too tight, decrease B.