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
where is the mass of the particle, is the component of the velocity perpendicular to the direction of the magnetic field, is the electric charge of the particle, and is the strength of the magnetic field.
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 is the speed of light in vacuum.
The above formula for the gyroradius also holds for relativistic motion when mass correction is considered. However, it must be remembered that the mass of the particle is the relativistic mass, rather than the rest mass. For calculations in accelerator and astroparticle physics, the formula for the gyroradius is rearranged to give the more practical expression
If the charged particle is moving, then it will experience a Lorentz force given by
where is the velocity vector and 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, , 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