Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 1

The Electromagnetic Field Equations

Physics — Landau & Lifshitz · relativistic-motion-parallel-em-fields

Problem

Determine the relativistic motion of a charge in parallel uniform electric and magnetic fields. Solution: The magnetic field has no influence on the motion along the common direction of

\mathbf{E}

and

\mathbf{H}

(the

z

axis), which therefore occurs under the influence of the electric field alone; therefore according to §20 we find:

z = \frac{\mathscr{E}_{\mathrm{kin}}}{eE}, \quad \mathscr{E}_{\mathrm{kin}} = \sqrt{\mathscr{E}_0^2 + (ceEt)^2}.

For the motion in the

xy

plane we have the equation

\dot{p}_x = \frac{e}{c}Hv_y, \quad \dot{p}_y = -\frac{e}{c}Hv_x,

\frac{d}{dt}(p_x + ip_y) = -i\frac{eH}{c}(v_x + iv_y) = -\frac{ieHc}{\mathscr{E}_{\mathrm{kin}}}(p_x + ip_y).

Consequently

p_x + ip_y = p_t e^{-i\phi}

, where

p_t

is the constant value of the projection of the momentum on the

xy

plane, and the auxiliary quantity

\phi

is defined by the relation

d\phi = eHc\, dt/\mathscr{E}_{\mathrm{kin}}

. The trajectory is a helix with radius

cp_t/eH

and monotonically increasing step, along which the particle moves with decreasing angular velocity

\dot{\phi} = eHc/\mathscr{E}_{\mathrm{kin}}

and with a velocity along the

z

axis which tends toward the value

c

Related theory

Electromagnetic Fields Forces And Fields

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