The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field.

$r_g = \frac{m v_{\perp}}{|q| B}$

where

• $r_g \$ is the gyroradius,
• $m \$ is the mass of the charged particle,
• $v_{\perp}$ is the velocity component perpendicular to the direction of the magnetic field,
• $q \$ is the charge of the particle, and
• $B \$ is the constant magnetic field.

(All units are in SI)

Similarly, the frequency of this circular motion is known as the gyrofrequency or cyclotron frequency, and is given in radian/second by:

$\omega_g = \frac{|q| B}{m}$

and in Hz by:

$\ f_g = \frac{q B}{2 \pi m}$

For electrons, this works out to be

$\nu_e = (2.8\times10^{10}\,\mathrm{Hz}/\mathrm{T})\times B$

## Relativistic case

The formula for the gyroradius also holds for relativistic motion. In that case, the velocity and mass of the moving object has to be replaced by the relativistic momentum $m v_{\perp} \rightarrow p_{\perp}$:

$r_g = \frac{p_{\perp}}{|q| B}$

For rule-of-thumb calculations in accelerator and astroparticle physics, the physical quantities can be expressed in proper units, which results in the simple numerical formula

$r_g/\mathrm{m} = 3.3 \times \frac{p_{\perp}/(\mathrm{GeV/c})}{|Z| (B/\mathrm{T})}$

where

• $Z \$ is the charge of the moving object in elementary units.

## Derivation

If the charged particle is moving, then it will experience a Lorentz force given by:

$\vec{F} = q(\vec{v} \times \vec{B})$

where $\vec{v}$ is the velocity vector, $\vec{B}$ is the magnetic field vector, and $q$ is the particle's electric charge.

Notice that the direction of the force is given by the cross product of the velocity and magnetic field. Thus, the Lorentz force will always act perpendicular to the direction of motion, causing the particle to move in a circle (gyrate). The radius of this circle $r_g$ can be determined by equating the magnitude of the Lorentz force to the centripetal force:

$\frac{m v_{\perp}^2}{r_g} = qv_{\perp}B$

where

$m$ is the particle mass (for high velocities the relativistic mass),
${v_{\perp}}$ is the velocity component perpendicular to the direction of the magnetic field, and
$B$ is the strength of the field.

Solving for $r_g$, the gyroradius is determined to be:

$r_g = \frac{m v_{\perp}}{q B}$

Thus, the gyroradius is directly proportional to the particle mass and velocity, and inversely proportional to the particle electric charge, and the magnetic field strength.