Larmor radius

The radius of the circle along which an electrically charged particle moves in a plane perpendicular to a magnetic field with magnetic induction $ \mathbf B $. The motion of the charge $ e $ in a uniform magnetic field takes place under the action of the Lorentz force and is described by the equation

$$ \tag{1 } \frac{\partial \mathbf p }{\partial t } = \ e [ \mathbf v , \mathbf B ] , $$

where $ \mathbf p $ is the momentum of the charged particle and $ \mathbf v $ is the velocity of the charge in the laboratory reference frame. The solution of (1) in a Cartesian coordinate system with the $ z $- axis directed along the field $ \mathbf B $ has the form

$$ \tag{2 } v _ {x} = v _ {0t} \cos ( \omega _ {L} t + \alpha ) ,\ \ v _ {y} = - v _ {0t} \sin ( \omega _ {L} t + \alpha ) ,\ \ $$

$$ v _ {z} = v _ {0z} , $$

$$ x = x _ {0} + r \sin ( \omega _ {L} t + \alpha ) ,\ y = y _ {0} + r \cos ( \omega _ {L} t + \alpha ) , $$

$$ z = z _ {0} + v _ {0z} t , $$

where $ \omega _ {L} = e c ^ {2} \mathbf B / \epsilon $ is the so-called Larmor frequency, $ \epsilon $ is the energy of the charged particle, which does not change under motion in a uniform magnetic field, $ v _ {0t} $, $ v _ {0z} $, $ \alpha $, $ x _ {0} $, $ y _ {0} $, $ z _ {0} $ are constants determined from the initial conditions, and

$$ r = \frac{v _ {0t} }{\omega _ {L} } = \ \frac{v _ {0t} \epsilon }{e c ^ {2} | \mathbf B | } $$

is the Larmor radius. In a uniform magnetic field the charge moves along a helix with axis along the magnetic field and Larmor radius $ r $. The velocity of the particle is constant.

If the velocity of the particle is small compared with the velocity of light, one can put approximately $ \epsilon = mc ^ {2} $ and the expression for the Larmor radius takes the form

$$ r = \frac{v _ {0t} }{\omega _ {0} } = \ \frac{v _ {0t} mc ^ {2} }{e | \mathbf B | } . $$

The magnetic moment of the system manifests itself as a result of the rotation of the charged particles in the magnetic field.

References

[1]	I.E. Tamm, "Fundamentals of the theory of electricity" , MIR (1979) (Translated from Russian)
[2]	L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1951) (Translated from Russian)
[a1]	P.C. Clemmow, J.P. Dougherty, "Electrodynamics of particles and plasmas" , Addison-Wesley (1969)