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The force that a given electromagnetic field exerts on a moving electrically-charged particle. An expression of the Lorentz force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608701.png" /> was first given by H.A. Lorentz (see [[#References|[1]]]):
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<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/l060/l060870/l0608702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608703.png" /> is the electric field strength, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608704.png" /> is the magnetic induction, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608705.png" /> is the velocity of the charged particle with respect to the coordinate system in which the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608708.png" /> are calculated, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l0608709.png" /> is the charge of the particle, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087010.png" /> is the velocity of light in vacuum. The expression for the Lorentz force is relativistically invariant (that is, it holds in any inertial reference system); it makes it possible to connect the equations for an electromagnetic field with the equations of motion of charged particles.
+
The force that a given electromagnetic field exerts on a moving electrically-charged particle. An expression of the Lorentz force  $  \mathbf F $
 +
was first given by H.A. Lorentz (see [[#References|[1]]]):
  
In a constant and uniform magnetic field the motion of a particle with mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087011.png" /> and charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087012.png" /> in a non-relativistic approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087013.png" /> is described by the equation
+
$$ \tag{1 }
 +
\mathbf F  = e \mathbf E +
 +
\frac{e}{c}
 +
[ \mathbf V , \mathbf B ] ,
 +
$$
  
<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/l060/l060870/l06087014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
where  $  \mathbf E $
 +
is the electric field strength,  $  \mathbf B $
 +
is the magnetic induction,  $  \mathbf V $
 +
is the velocity of the charged particle with respect to the coordinate system in which the quantities  $  \mathbf E $,
 +
$  \mathbf B $,
 +
$  \mathbf F $
 +
are calculated,  $  e $
 +
is the charge of the particle, and  $  c $
 +
is the velocity of light in vacuum. The expression for the Lorentz force is relativistically invariant (that is, it holds in any inertial reference system); it makes it possible to connect the equations for an electromagnetic field with the equations of motion of charged particles.
  
In a rectangular coordinate system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087015.png" />-axis directed along the outward magnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087016.png" />, the solution of (2) has the form
+
In a constant and uniform magnetic field the motion of a particle with mass  $  m $
 +
and charge  $  e $
 +
in a non-relativistic approximation  $  ( \mathbf V \ll  c ) $
 +
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/l060/l060870/l06087017.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
m
 +
\frac{d \mathbf V }{dt}
 +
  =
 +
\frac{e}{c}
 +
[ \mathbf V , \mathbf B ] .
 +
$$
  
<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/l060/l060870/l06087018.png" /></td> </tr></table>
+
In a rectangular coordinate system with  $  z $-
 +
axis directed along the outward magnetic field  $  B $,
 +
the solution of (2) has the form
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087019.png" /> is the Larmor frequency of rotation of the particle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087020.png" /> is the radius of rotation of the particle (the [[Larmor radius|Larmor radius]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087021.png" /> is the initial phase of the rotation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087022.png" /> is the initial velocity of the particle. Thus, in a uniform magnetic field the charge moves along a helix with axis along the magnetic field.
+
$$
 +
= x _ {0} + r  \sin ( \omega _ {L} t + \alpha ) ,\ \
 +
= y _ {0} + r  \cos ( \omega _ {L} t + \alpha ) ,
 +
$$
  
If the electric field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087023.png" /> is not equal to zero, the motion has a more complicated character. There occurs a displacement of the centre of rotation of the particle across the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087024.png" /> (so-called drift). The mean value of drift in vector form is
+
$$
 +
= z _ {0} + \mathbf V _ {0z} t ,
 +
$$
  
<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/l060/l060870/l06087025.png" /></td> </tr></table>
+
where  $  \omega _ {L} = e | \mathbf B | / m c $
 +
is the Larmor frequency of rotation of the particle,  $  r = | \mathbf V _ {0t} | / \omega _ {L} $
 +
is the radius of rotation of the particle (the [[Larmor radius|Larmor radius]]),  $  \alpha $
 +
is the initial phase of the rotation, and  $  \mathbf V _ {0} $
 +
is the initial velocity of the particle. Thus, in a uniform magnetic field the charge moves along a helix with axis along the magnetic field.
  
The unaveraged motion of the particle in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060870/l06087026.png" />-plane in this case takes place along a trochoid.
+
If the electric field  $  \mathbf E $
 +
is not equal to zero, the motion has a more complicated character. There occurs a displacement of the centre of rotation of the particle across the field  $  \mathbf B $(
 +
so-called drift). The mean value of drift in vector form is
 +
 
 +
$$
 +
\mathbf V  =  c
 +
 
 +
\frac{[ \mathbf E , \mathbf B ] }{| \mathbf B |  ^ {2} }
 +
.
 +
$$
 +
 
 +
The unaveraged motion of the particle in the $  xy $-
 +
plane in this case takes place along a trochoid.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.A. Lorentz,  "The theory of electrons and its applications to the phenomena of light and radiant heat" , Teubner  (1909)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "The classical theory of fields" , Pergamon  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.A. Lorentz,  "The theory of electrons and its applications to the phenomena of light and radiant heat" , Teubner  (1909)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "The classical theory of fields" , Pergamon  (1975)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.G. Levich,  "Theoretical physics" , '''1. Theory of the electromagnetic field''' , North-Holland  (1970)  pp. 6; 364; 366</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.A. Hylleraas,  "Mathematical and theoretical physics" , '''2''' , Wiley (Interscience)  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.C. Clemmow,  J.P. Dougherty,  "Electrodynamics of particles and plasmas" , Addison-Wesley  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.G. Levich,  "Theoretical physics" , '''1. Theory of the electromagnetic field''' , North-Holland  (1970)  pp. 6; 364; 366</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.A. Hylleraas,  "Mathematical and theoretical physics" , '''2''' , Wiley (Interscience)  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.C. Clemmow,  J.P. Dougherty,  "Electrodynamics of particles and plasmas" , Addison-Wesley  (1969)</TD></TR></table>

Revision as of 04:11, 6 June 2020


The force that a given electromagnetic field exerts on a moving electrically-charged particle. An expression of the Lorentz force $ \mathbf F $ was first given by H.A. Lorentz (see [1]):

$$ \tag{1 } \mathbf F = e \mathbf E + \frac{e}{c} [ \mathbf V , \mathbf B ] , $$

where $ \mathbf E $ is the electric field strength, $ \mathbf B $ is the magnetic induction, $ \mathbf V $ is the velocity of the charged particle with respect to the coordinate system in which the quantities $ \mathbf E $, $ \mathbf B $, $ \mathbf F $ are calculated, $ e $ is the charge of the particle, and $ c $ is the velocity of light in vacuum. The expression for the Lorentz force is relativistically invariant (that is, it holds in any inertial reference system); it makes it possible to connect the equations for an electromagnetic field with the equations of motion of charged particles.

In a constant and uniform magnetic field the motion of a particle with mass $ m $ and charge $ e $ in a non-relativistic approximation $ ( \mathbf V \ll c ) $ is described by the equation

$$ \tag{2 } m \frac{d \mathbf V }{dt} = \frac{e}{c} [ \mathbf V , \mathbf B ] . $$

In a rectangular coordinate system with $ z $- axis directed along the outward magnetic field $ B $, the solution of (2) has the form

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

$$ z = z _ {0} + \mathbf V _ {0z} t , $$

where $ \omega _ {L} = e | \mathbf B | / m c $ is the Larmor frequency of rotation of the particle, $ r = | \mathbf V _ {0t} | / \omega _ {L} $ is the radius of rotation of the particle (the Larmor radius), $ \alpha $ is the initial phase of the rotation, and $ \mathbf V _ {0} $ is the initial velocity of the particle. Thus, in a uniform magnetic field the charge moves along a helix with axis along the magnetic field.

If the electric field $ \mathbf E $ is not equal to zero, the motion has a more complicated character. There occurs a displacement of the centre of rotation of the particle across the field $ \mathbf B $( so-called drift). The mean value of drift in vector form is

$$ \mathbf V = c \frac{[ \mathbf E , \mathbf B ] }{| \mathbf B | ^ {2} } . $$

The unaveraged motion of the particle in the $ xy $- plane in this case takes place along a trochoid.

References

[1] H.A. Lorentz, "The theory of electrons and its applications to the phenomena of light and radiant heat" , Teubner (1909)
[2] L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Pergamon (1975) (Translated from Russian)

Comments

References

[a1] B.G. Levich, "Theoretical physics" , 1. Theory of the electromagnetic field , North-Holland (1970) pp. 6; 364; 366
[a2] E.A. Hylleraas, "Mathematical and theoretical physics" , 2 , Wiley (Interscience) (1970)
[a3] P.C. Clemmow, J.P. Dougherty, "Electrodynamics of particles and plasmas" , Addison-Wesley (1969)
How to Cite This Entry:
Lorentz force. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lorentz_force&oldid=11653
This article was adapted from an original article by V.V. Parail (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article