View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575901.png" />. The motion of the charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575902.png" /> in a uniform magnetic field takes place under the action of the Lorentz force and is described by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575904.png" /> is the momentum of the charged particle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575905.png" /> is the velocity of the charge in the laboratory reference frame. The solution of (1) in a Cartesian coordinate system with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575906.png" />-axis directed along the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575907.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l0575909.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759012.png" /> is the so-called Larmor frequency, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759013.png" /> is the energy of the charged particle, which does not change under motion in a uniform magnetic field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759019.png" /> are constants determined from the initial conditions, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759020.png" /></td> </tr></table>

is the Larmor radius. In a uniform magnetic field the charge moves along a helix with axis along the magnetic field and Larmor radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759021.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759022.png" /> and the expression for the Larmor radius takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057590/l05759023.png" /></td> </tr></table>

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.E. Tamm,   "Fundamentals of the theory of electricity" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshitz,   "The classical theory of fields" , Addison-Wesley  (1951)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.C. Clemmow,   J.P. Dougherty,   "Electrodynamics of particles and plasmas" , Addison-Wesley  (1969)</TD></TR></table>