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| 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]].
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| :<math>r_g = \frac{m v_{\perp}}{|q| B}</math>
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| where
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| :*<math>r_g \ </math> is the gyroradius,
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| :*<math>m \ </math> is the mass of the charged particle,
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| :*<math>v_{\perp}</math> is the velocity component perpendicular to the direction of the magnetic field,
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| :*<math>q \ </math> is the charge of the particle, and
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| :*<math>B \ </math> is the constant magnetic field.
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| (All units are in [[SI]])
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| Similarly, the [[frequency]] of this circular motion is known as the '''gyrofrequency''' or '''[[cyclotron frequency]]''', and is given in radian/second by:
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| :<math>\omega_g = \frac{|q| B}{m}</math>
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| and in Hz by:
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| :<math>\ f_g = \frac{q B}{2 \pi m}</math>
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| For electrons, this works out to be
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| :<math>\nu_e = (2.8\times10^{10}\,\mathrm{Hz}/\mathrm{T})\times B</math>
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| ==Relativistic case==
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| The formula for the gyroradius also holds for [[Special_relativity|relativistic motion]]. In that case, the velocity and mass of the moving object has to be replaced by the relativistic momentum <math>m v_{\perp} \rightarrow p_{\perp}</math>:
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| <math>r_g = \frac{p_{\perp}}{|q| B}</math>
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| For rule-of-thumb calculations in [[Particle_accelerator|accelerator]] and [[Astroparticle_physics|astroparticle]] physics, the physical quantities can be expressed in proper units, which results in the simple numerical formula
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| :<math>r_g/\mathrm{m} = 3.3 \times \frac{p_{\perp}/(\mathrm{GeV/c})}{|Z| (B/\mathrm{T})}</math>
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| where
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| :*<math>Z \ </math> is the charge of the moving object in elementary units.
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| ==Derivation==
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| If the charged particle is moving, then it will experience a [[Lorentz force]] given by:
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| :<math>\vec{F} = q(\vec{v} \times \vec{B})</math>
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| where <math>\vec{v}</math> is the velocity vector, <math>\vec{B}</math> is the magnetic field vector, and <math>q</math> is the particle's [[electric charge]].
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| 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 move in a circle ([[gyration|gyrate]]). The radius of this circle <math>r_g</math> can be determined by equating the magnitude of the Lorentz force to the [[centripetal force]]:
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| :<math>\frac{m v_{\perp}^2}{r_g} = qv_{\perp}B</math> | |
| where
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| :<math>m</math> is the particle [[mass]] (for high velocities the [[relativistic mass]]), | |
| :<math>{v_{\perp}}</math> is the velocity component perpendicular to the direction of the magnetic field, and | |
| :<math>B</math> is the strength of the field.
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| Solving for <math>r_g</math>, the gyroradius is determined to be:
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| :<math>r_g = \frac{m v_{\perp}}{q B}</math>
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| Thus, the gyroradius is directly proportional to the particle mass and velocity, and inversely proportional to the particle electric charge, and the magnetic field strength.
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| ==See also==
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| * [[Cyclotron]]
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| * [[Magnetosphere particle motion]]
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| ==References & further reading==
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| <div class="references-small">
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| # {{cite book | first=Francis F. | last=Chen | title=Introduction to Plasma Physics and Controlled Fusion, Vol. 1: Plasma Physics, 2nd ed. | publisher=Plenum Press | location=New York, NY USA | year=1984 | isbn=0-306-41332-9}}
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| </div>
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| [[Category:Plasma physics]]
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| [[Category:Accelerator physics]]
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Hello. Let me introduce the writer. Her title is Emilia Shroyer but it's not the most feminine name out there. North Dakota is our beginning place. One of the very very best issues in the world for me is to do aerobics and now I'm trying to earn money with it. Hiring is his profession.
Feel free to surf to my weblog :: nuvem.tk