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Concept
Gyroradius
Summary
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 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.
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.
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 , that is the velocity of light, because the magnetic field is expressed in units .
For relativistic particles the classical equation needs to be interpreted in terms of particle momentum :
where 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 is the speed of light, is the unit of Giga-electronVolts, and is the elementary charge.
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.
Official source
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