Gyroradius

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

$r_{g} = \frac{m v_{\perp}}{|q| B}$

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

$\omega_{g} = \frac{|q| B}{m}$

in units of radians/second.^[1]

Variants

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

$\Omega_{g} = \frac{q B}{m}$

or express it in units of Hertz with

$f_{g} = \frac{q B}{2 \pi m}$.

For electrons, this frequency can be reduced to

$f_{g,e} = (2.8\times10^{10}\,\mathrm{Hertz}/\mathrm{Tesla})\times B$.

In cgs units, the gyroradius is given by

$r_{g} = \frac{m c v_{\perp}}{|q| B}$

and the gyrofrequency is

$\omega_{g} = \frac{|q| B}{m c}$,

where $c$ is the speed of light in vacuum.

Relativistic case

The above formula for the gyroradius also holds for relativistic motion when mass correction is considered. However, it must be remembered that the mass of the particle is the relativistic mass, rather than the rest mass. For calculations in accelerator and astroparticle physics, the formula for the gyroradius is rearranged to give the more practical expression

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

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

Derivation

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

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

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

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

Rearranging, the gyroradius can be expressed as

$r_{g} = \frac{m v_{\perp}}{|q| B}$.

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

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

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

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

and therefore

$\omega_{g} = \frac{|q| B}{m}$.

See also

• Beam rigidity
• Cyclotron frequency
• Cyclotron
• Magnetosphere particle motion
• Gyrokinetics

References

1. 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.