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.[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]

VariantsEdit

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  .

Relativistic caseEdit

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.

DerivationEdit

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

 

and therefore

 .

See alsoEdit

ReferencesEdit

  1. ^ a b Chen, Francis F. (1983). Introduction to Plasma Physics and Controlled Fusion, Vol. 1: Plasma Physics, 2nd ed. New York, NY USA: Plenum Press. p. 20. ISBN 978-0-306-41332-2.