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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. In SI units, the non-relativistic gyroradius is given by

where ${\displaystyle m}$ is the mass of the particle, ${\displaystyle v_{\perp }}$ is the component of the velocity perpendicular to the direction of the magnetic field, ${\displaystyle q}$ is the electric charge of the particle, and ${\displaystyle B}$ is the magnetic field flux density.^[1]

The angular frequency of this circular motion is known as the gyrofrequency, or cyclotron frequency, and can be expressed as

in units of radians/second.^[1]

Variants

It is often useful to give the gyrofrequency a sign with the definition

or express it in units of hertz with For electrons, this frequency can be reduced to

In cgs-units the gyroradius

and the corresponding gyrofrequency include a factor ${\displaystyle c}$, that is the velocity of light, because the magnetic field is expressed in units ${\displaystyle =\mathrm {g^{1/2}cm^{-1/2}s^{-1}} }$.

Relativistic case

For relativistic particles the classical equation needs to be interpreted in terms of particle momentum ${\displaystyle p=\gamma mv}$:

where ${\displaystyle \gamma }$ is the Lorentz factor. This equation is correct also in the non-relativistic case.

For calculations in accelerator and astroparticle physics, the formula for the gyroradius can be rearranged to give

where ${\displaystyle c}$ is the speed of light, ${\displaystyle \mathrm {GeV} }$ is the unit of Giga-electronVolts, and ${\displaystyle e}$ is the elementary charge.

Derivation

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

where ${\displaystyle {\vec {v}}}$ is the velocity vector and ${\displaystyle {\vec {B}}}$ is the magnetic field vector.

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 gyrate, or move in a circle. The radius of this circle, ${\displaystyle r_{g}}$, can be determined by equating the magnitude of the Lorentz force to the centripetal force as

Rearranging, the gyroradius can be expressed as Thus, the gyroradius is directly proportional to the particle mass and perpendicular velocity, while it is inversely proportional to the particle electric charge and the magnetic field strength. The time it takes the particle to complete one revolution, called the period, can be calculated to be Since the period is the reciprocal of the frequency we have found