Classical Electrodynamics · Problem 12.6

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · motion-combined-electric-magnetic-fields

Problem

Static, uniform electric and magnetic fields,

\mathbf{E}

and

\mathbf{B}

, make an angle of

\theta

with respect to each other. (a) By a suitable choice of axes, solve the force equation for the motion of a particle of charge

e

and mass

m

in rectangular coordinates. (b) For

\mathbf{E}

and

\mathbf{B}

parallel, show that with appropriate constants of integration, etc., the parametric solution can be written

\begin{aligned} x &= AR\sin\phi, \quad y = AR\cos\phi, \quad z = \frac{R}{\rho}\sqrt{1+A^2}\cosh(\rho\phi)\\ ct &= \frac{R}{\rho}\sqrt{1+A^2}\sinh(\rho\phi) \end{aligned}

where

R = (mc^2/eB)

\rho = (E/B)

A

is an arbitrary constant, and

\phi

is the parameter [actually

c/R

times the proper time].