Lorentz force

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

(1)

where is the electric field strength, is the magnetic induction, is the velocity of the charged particle with respect to the coordinate system in which the quantities , , are calculated, is the charge of the particle, and 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 and charge in a non-relativistic approximation is described by the equation

(2)

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

where is the Larmor frequency of rotation of the particle, is the radius of rotation of the particle (the Larmor radius), is the initial phase of the rotation, and 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 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 (so-called drift). The mean value of drift in vector form is

The unaveraged motion of the particle in the -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)

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)