Larmor Radius Calculator

Reviewed by: JJ Ben-Joseph

Understanding the Larmor Radius

The Larmor radius, also called the gyroradius, describes the circular path taken by a charged particle moving perpendicular to a uniform magnetic field. When an electron, proton, or any ion enters a magnetic field, the Lorentz force continuously deflects it at right angles to both its velocity and the field. The particle therefore traces a circle, and the radius of that circle is a key scale in plasma physics, astrophysics, and accelerator design. Mathematically, the radius is given by $r=\frac{mv}{\left|q\right|B}$, where $m$ is mass, $v$ velocity perpendicular to the field, $\left|q\right|$ charge magnitude, and $B$ the magnetic field strength.

Although the formula is compact, its implications are broad. In Earth’s magnetosphere, for instance, the gyroradius of a 10 keV electron in a 50 nT field is only tens of kilometers, far smaller than planetary scales, justifying models that treat particle motion as tied to magnetic field lines. In fusion devices like tokamaks, designing magnetic confinement requires balancing particle energy, mass, and field strength to keep gyroradii tiny compared to the machine size. Cosmic-ray physicists likewise estimate gyroradii to gauge whether magnetic fields in interstellar space can trap high‑energy particles or allow them to wander between stars.

The calculator on this page lets you solve for any one of the five variables in the gyroradius relation. Leave exactly one input blank, fill in the other four, and the script computes the missing quantity. If more than one field is empty, or if all are filled, the calculator reports an error. This flexibility assists in a variety of tasks: choosing magnet strength for a particle beam line, estimating the velocity of ions detected in space plasmas, or assessing the charge‑to‑mass ratio of an unknown particle from observed curvature in a bubble chamber photograph.

The gyroradius concept originates from the early study of charged particle motion. In 1898, physicist Joseph Larmor described how electrons orbit magnetic field lines, giving rise to the term “Larmor radius.” His ideas laid groundwork for later developments in cyclotrons and synchrotrons—machines that accelerate particles using magnetic fields. In these devices, the gyroradius determines the orbit size; increasing the magnetic field shrinks the orbit, allowing higher energy particles to be confined within a reasonable footprint.

To illustrate the scale of gyroradii, consider the table below. It lists typical values for electrons and protons in representative magnetic fields and velocities found in laboratory and space environments.

These examples show how profoundly mass and magnetic field influence the orbit. Light electrons spiral tightly even at moderate fields, while heavy protons require either higher fields or lower speeds to achieve the same confinement. In astrophysical contexts, where fields can be extremely weak, gyroradii may reach astronomical scales. Cosmic rays traversing the Galaxy with energies near 10¹⁵ eV can have gyroradii comparable to the size of spiral arms, allowing them to diffuse across vast distances.

In plasma physics, the gyroradius sets a natural length scale that separates “micro” from “macro” behavior. When variations in fields or densities occur over distances much larger than the gyroradius, the particles complete many orbits before noticing the change, and their motion can be averaged to derive fluid equations. When perturbations are comparable to or smaller than the gyroradius, particles experience them directly, leading to kinetic effects like wave–particle interactions or cyclotron resonances. Understanding this scale is therefore essential for interpreting satellite data, diagnosing laboratory plasmas, or modeling magnetohydrodynamic turbulence.

The Larmor radius also determines how charged particles respond to collisions. In weakly collisional plasmas, a particle may circle thousands of times before colliding with another, making the magnetic field the dominant influence on its trajectory. In denser media where collisions occur frequently, the gyroradius may be comparable to the mean free path, leading to diffusive motion rather than coherent orbits. Engineers designing magnetic refrigeration systems, mass spectrometers, or particle traps must weigh these effects when choosing operating conditions.

Mathematically, the gyroradius can be derived from the Lorentz force law $\backslash vecF=q(\backslash vecv\backslash times\backslash vecB)$. For motion perpendicular to the field, the force magnitude is $F=\left|q\right|\mathrm{vB}$. Setting this equal to the centripetal force $F=\frac{m}{\mathrm{v^2}}r$ and solving for $r$ yields the formula implemented in the calculator. If the particle’s velocity is not perpendicular, only the perpendicular component contributes to circular motion while the parallel component leads to a helix.

In educational settings, exploring the Larmor radius deepens understanding of magnetism and motion. Students can vary parameters to see how each influences the result, reinforcing proportional reasoning: doubling velocity doubles the radius, doubling field strength halves it, and increasing charge magnitude or mass adjusts the radius inversely or directly. Real-world problems—such as estimating electron paths in cathode-ray tubes or predicting ion trajectories in mass spectrometers—become more tangible when this fundamental relation is at hand.

Although the classic formula assumes nonrelativistic speeds, the concept extends into relativity by replacing mass with relativistic mass or, more appropriately, by using momentum $p$. The general expression becomes $r=\frac{p}{\left|}$. High-energy physics experiments routinely apply this version when reconstructing particle tracks in detectors. The curvature of charged particle trajectories through magnetic fields reveals their momentum, allowing physicists to identify particles and measure their energies.

Beyond individual particle motion, the gyroradius plays a role in macroscopic phenomena like the Hall effect or magnetoresistance, where charge carriers deflect under magnetic fields. In semiconductor physics, understanding how electron cyclotron orbits compare to impurity spacing or device dimensions informs the design of sensors and transistors. In geophysics, analyses of cosmic ray penetration or auroral particle precipitation hinge on gyroradius considerations to determine how particles access the atmosphere.

The notion of gyrofrequency accompanies the gyroradius. The angular frequency of rotation is $\mathrm{\backslash omega\_c}=\frac{\left|}{q}m$, and the period is $T=\frac{2}{\mathrm{\backslash pi}}\mathrm{\backslash omega\_c}$. These quantities together with the radius describe the full helical motion of particles. The calculator focuses on radius, but with the same inputs you can easily compute frequency or period to explore how quickly particles whirl around field lines.

The Larmor radius is thus a cornerstone concept bridging microscopic particle behavior and large-scale magnetic structures. By providing a simple, interactive tool alongside a detailed explanation, this page aims to make the physics accessible for students, hobbyists, and professionals alike. Experiment with the inputs, compare to the table, and consider how this tiny circle governs dynamics from laboratory plasmas to the vast interstellar medium.

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