**Radiation from an Accelerated
Charge**

[see also Longair, High Energy
Physics, 2nd edition, vol. 1, p. 64, and
Wikipedia]

Maxwell's equations imply that all
classical electromagnetic
radiation is
ultimately generated by accelerating electrical charges. It is possible
to derive the intensity and angular distribution of the radiation from
a point charge (a charged particle) subject to an arbitrary but small
acceleration $\Delta v / \Delta t$ via Maxwell's equations (and retarded
potentials), but the
complicated math obscures the physical interpretation that remains
clear in J. J. Thomson's simpler derivation.

If a particle with electrical charge
$q$ is at rest, Coulomb's law
implies that its electric field lines will be purely radial: $E =
E_{\rm r}$.
Suppose a charged particle is accelerated to some small velocity
$\Delta v \ll
c$ in some short time $\Delta t$. At time $t$ later, there will
be a perpendicular component of electric field

$${E_{\bot} \over
E_{\rm r}} = {\Delta v \, t \, \sin\theta \over c \Delta t}~,$$

where
$\theta$ is the angle between the acceleration vector and the line from
the charge to the observer.