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 non-relativistic 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 magnetic field flux density.

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.

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 $$r_{g} = \frac{m c v_{\perp}}{|q| B}$$ and the corresponding gyrofrequency $$\omega_{g} = \frac{|q| B}{m c}$$ include a factor $$ c $$, that is the velocity of light, because the magnetic field is expressed in units $$ [B] = \mathrm{g^{1/2} cm^{-1/2} s^{-1}} $$.

Relativistic case
For relativistic particles the classical equation needs to be interpreted in terms of particle momentum $$p=\gamma m v$$: $$r_{g} = \frac{p_{\perp}} = \frac{\gamma m v_{\perp}}{|q| B}$$ where $$\gamma$$ 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 give $$r_{g}/\mathrm{meter} = 3.3 \times \frac{( \gamma m c^{2}/\mathrm{GeV})(v_{\perp} / c)}{(|q|/e) (B/\mathrm{Tesla})},$$ 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 $$\omega_{g} = \frac{|q| B}{m}.$$