Larmor radius

Orbit of a negatively charged particle (e.g. electron) with Larmor radius;
the magnetic field runs perpendicular into the plane of the drawing ${\displaystyle {\vec {B}}}$

The Larmor radius (after Joseph Larmor ; due to its meaning in the cyclotron, also cyclotron radius ; other designation gyroradius / gyration radius ) is the radius of the circular motion of a charged particle in a homogeneous magnetic field : ${\displaystyle r_{g}\,}$

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

With

• ${\displaystyle m\ }$ Mass of the charged particle
• ${\displaystyle v_{\perp }}$ Velocity component perpendicular to the magnetic field lines
• ${\displaystyle q\ }$ electrical charge of the particle
• ${\displaystyle B\ }$ magnetic flux density of the homogeneous magnetic field.

The frequency of this circular movement is called the cyclotron frequency or gyration frequency:

${\displaystyle \nu ={\frac {q\cdot B}{2\cdot \pi \cdot m}}}$

It is to be distinguished from the Larmor frequency , which describes the frequency of the spin precession .

The size is also called magnetic stiffness . ${\displaystyle B\cdot r_{g}}$

Derivation

The Lorentz force acts on a charged particle moving in a magnetic field :

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

With

• ${\displaystyle {\vec {v}}}$ Velocity vector of the particle,
• ${\displaystyle {\vec {B}}}$ Magnetic flux density vector .

The direction of the force is determined by the cross product of the speed and the magnetic flux density. The Lorentz force therefore always acts perpendicular to the direction of movement and forces the particle, provided the magnetic field is the same (homogeneous) everywhere, onto a circular path.

Equating Lorentz force and centripetal force :

${\displaystyle q\cdot v_{\perp }\cdot B={\frac {m\cdot v_{\perp }^{2}}{r_{g}}}}$

results by solving for the above. Formula for the radius of the circular motion. ${\displaystyle r_{g}}$

Normalized gyro radius

In nuclear fusion technology, the Larmor radius based on a typical expansion of the plasma (for toroidal geometries , the small radius a is used) is called the normalized gyroradius :

${\displaystyle \rho ^{*}={\frac {r_{g}}{a}}.}$

It is an important dimensionless parameter for the dimensional analysis of fusion reactors.

literature

• Ulrich Stroth: Plasma Physics: Phenomena, Fundamentals, Applications . Vieweg + Teubner, 2011, ISBN 978-3-8348-1615-3 , pp. 15 ( limited preview in Google Book search).