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
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
^ abChen, Francis F. (1983). Introduction to Plasma Physics and Controlled Fusion, Vol. 1: Plasma Physics, 2nd ed. New York, NY USA: Plenum Press. p. 20. ISBN978-0-306-41332-2.
March 17, 2024
gyroradius, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, december, 2023,. This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Gyroradius news newspapers books scholar JSTOR December 2023 Learn how and when to remove this template message 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 byr g m v q B displaystyle r g frac mv perp q B where m displaystyle m is the mass of the particle v displaystyle v perp is the component of the velocity perpendicular to the direction of the magnetic field q displaystyle q is the electric charge of the particle and B displaystyle B is the magnetic field flux density 1 The angular frequency of this circular motion is known as the gyrofrequency or cyclotron frequency and can be expressed asw g q B m displaystyle omega g frac q B m in units of radians second 1 Contents 1 Variants 2 Relativistic case 3 Derivation 4 See also 5 ReferencesVariants editIt is often useful to give the gyrofrequency a sign with the definitionw g q B m displaystyle omega g frac qB m nbsp or express it in units of hertz with f g q B 2 p m displaystyle f g frac qB 2 pi m nbsp For electrons this frequency can be reduced to f g e 2 8 10 10 h e r t z t e s l a B displaystyle f g e 2 8 times 10 10 mathrm hertz mathrm tesla times B nbsp In cgs units the gyroradiusr g m c v q B displaystyle r g frac mcv perp q B nbsp and the corresponding gyrofrequency w g q B m c displaystyle omega g frac q B mc nbsp include a factor c displaystyle c nbsp that is the velocity of light because the magnetic field is expressed in units B g 1 2 c m 1 2 s 1 displaystyle B mathrm g 1 2 cm 1 2 s 1 nbsp Relativistic case editFor relativistic particles the classical equation needs to be interpreted in terms of particle momentum p g m v displaystyle p gamma mv nbsp r g p q B g m v q B displaystyle r g frac p perp q B frac gamma mv perp q B nbsp where g displaystyle gamma nbsp 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 giver g m e t e r 3 3 g m c 2 G e V v c q e B T e s l a displaystyle r g mathrm meter 3 3 times frac gamma mc 2 mathrm GeV v perp c q e B mathrm Tesla nbsp where c displaystyle c nbsp is the speed of light G e V displaystyle mathrm GeV nbsp is the unit of Giga electronVolts and e displaystyle e nbsp is the elementary charge Derivation editIf the charged particle is moving then it will experience a Lorentz force given byF q v B displaystyle vec F q vec v times vec B nbsp where v displaystyle vec v nbsp is the velocity vector and B displaystyle vec B nbsp 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 r g displaystyle r g nbsp can be determined by equating the magnitude of the Lorentz force to the centripetal force asm v 2 r g q v B displaystyle frac mv perp 2 r g q v perp B nbsp Rearranging the gyroradius can be expressed as r g m v q B displaystyle r g frac mv perp q B nbsp 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 T g 2 p r g v displaystyle T g frac 2 pi r g v perp nbsp Since the period is the reciprocal of the frequency we have found f g 1 T g q B 2 p m displaystyle f g frac 1 T g frac q B 2 pi m nbsp and therefore w g q B m displaystyle omega g frac q B m nbsp See also editBeam rigidity Cyclotron Magnetosphere particle motion GyrokineticsReferences edit 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 Retrieved from https en wikipedia org w index php title Gyroradius amp oldid 1187782771, wikipedia, wiki, book, books, library,