Gyroradius

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 formula_2 is the mass of the particle, formula_3 is the component of the velocity perpendicular to the direction of the magnetic field, formula_4 is the electric charge of the particle, and formula_5 is the strength of the magnetic field.
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.
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 formula_12, that is the velocity of light, because the magnetic field is expressed in units formula_13.
Relativistic case.
For relativistic particles the classical equation needs to be interpreted in terms of particle momentum formula_14:
where formula_16 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 formula_18 is the speed of light, formula_19 is the unit of Giga-electronVolts, and formula_20 is the elementary charge.
Derivation.
If the charged particle is moving, then it will experience a Lorentz force given by
where formula_22 is the velocity vector and formula_23 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, formula_24, 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
and therefore