# Gyroradius

VIDEOS 1 TO 50

GO TO RESULTS [51 .. 100]

### WIKIPEDIA ARTICLE

Jump to: navigation, search

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

${\displaystyle r_{g}={\frac {mv_{\perp }}{|q|B}}}$

where ${\displaystyle m}$ is the mass of the particle, ${\displaystyle v_{\perp }}$ is the component of the velocity perpendicular to the direction of the magnetic field, ${\displaystyle q}$ is the electric charge of the particle, and ${\displaystyle B}$ 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

${\displaystyle \omega _{g}={\frac {|q|B}{m}}}$

in units of radians/second.[1]

## Variants

It is often useful to give the gyrofrequency a sign with the definition

${\displaystyle \Omega _{g}={\frac {qB}{m}}}$

or express it in units of Hertz with

${\displaystyle f_{g}={\frac {qB}{2\pi m}}}$.

For electrons, this frequency can be reduced to

${\displaystyle f_{g,e}=(2.8\times 10^{10}\,\mathrm {Hertz} /\mathrm {Tesla} )\times B}$.

In cgs units, the gyroradius is given by

${\displaystyle r_{g}={\frac {mcv_{\perp }}{|q|B}}}$

and the gyrofrequency is

${\displaystyle \omega _{g}={\frac {|q|B}{mc}}}$,

where ${\displaystyle c}$ is the speed of light in vacuum.

## Relativistic case

For relativistic particles the Lorentz factor must be included:

${\displaystyle r_{g}={\frac {\gamma mv_{\perp }}{|q|B}}}$.

For calculations in accelerator and astroparticle physics, the formula for the gyroradius can be rearranged to give

${\displaystyle r_{g}/\mathrm {meter} =3.3\times {\frac {(\gamma mc^{2}/\mathrm {GeV} )(v_{\perp }/c)}{(|q|/e)(B/\mathrm {Tesla} )}}}$,

where ${\displaystyle c}$ is the speed of light, ${\displaystyle \mathrm {GeV} }$ is the unit of Giga-electronVolts, and ${\displaystyle e}$ is the elementary charge.

## Derivation

If the charged particle is moving, then it will experience a Lorentz force given by

${\displaystyle {\vec {F}}=q({\vec {v}}\times {\vec {B}})}$,

where ${\displaystyle {\vec {v}}}$ is the velocity vector and ${\displaystyle {\vec {B}}}$ 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, ${\displaystyle r_{g}}$, can be determined by equating the magnitude of the Lorentz force to the centripetal force as

${\displaystyle {\frac {mv_{\perp }^{2}}{r_{g}}}=|q|v_{\perp }B}$.

Rearranging, the gyroradius can be expressed as

${\displaystyle r_{g}={\frac {mv_{\perp }}{|q|B}}}$.

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

${\displaystyle T_{g}={\frac {2\pi r_{g}}{v_{\perp }}}}$.

Since the period is the reciprocal of the frequency we have found

${\displaystyle f_{g}={\frac {1}{T_{g}}}={\frac {|q|B}{2\pi m}}}$

and therefore

${\displaystyle \omega _{g}={\frac {|q|B}{m}}}$.

## References

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 0-306-41332-9.

### Disclaimer

None of the audio/visual content is hosted on this site. All media is embedded from other sites such as GoogleVideo, Wikipedia, YouTube etc. Therefore, this site has no control over the copyright issues of the streaming media.

All issues concerning copyright violations should be aimed at the sites hosting the material. This site does not host any of the streaming media and the owner has not uploaded any of the material to the video hosting servers. Anyone can find the same content on Google Video or YouTube by themselves.

The owner of this site cannot know which documentaries are in public domain, which has been uploaded to e.g. YouTube by the owner and which has been uploaded without permission. The copyright owner must contact the source if he wants his material off the Internet completely.

Wikipedia content is licensed under the GFDL and (CC) license