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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]]),
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| :<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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