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The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) defines 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.

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

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

For electrons, this works out to be

$\nu_e = (2.80\times10^{10}\,\mathrm{Hz})\times(B/\mathrm{T})$

## 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 rg 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,
${v_{\perp}}$ is the velocity component perpendicular to the direction of the magnetic field, and
B is the strength of the field.

Solving for rg, 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.