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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]]]):

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

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|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====
<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>
<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>