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Difference between revisions of "Lorentz force"

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====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>
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<table>
 
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<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>
====Comments====
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<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>
 
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<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>
====References====
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> E.A. Hylleraas, "Mathematical and theoretical physics" , '''2''' , Wiley (Interscience)  (1970)</TD></TR>
<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>
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<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>
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</table>

Latest revision as of 18:01, 27 May 2024


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)
How to Cite This Entry:
Lorentz force. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lorentz_force&oldid=55800
This article was adapted from an original article by V.V. Parail (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article