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 gyroradius is given by
Contents
where
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 is given by
and the gyrofrequency is
where
Relativistic case
The above formula for the gyroradius also holds for relativistic motion when mass correction is considered. For calculations in accelerator and astroparticle physics, the formula for the gyroradius is rearranged to give the more practical expression
where
Derivation
If the charged particle is moving, then it will experience a Lorentz force given by
where
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,
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