Lorentz force

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)
[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)