We begin with the Liénard–Wiechert potentials for a point particle with charge
in arbitrary motion:
The electric field has two terms, one determined by the velocity
and other by the acceleration
, respectively proportional to
. Because at large distances the acceleration terms dominate, these are also known as radiation fields. The Poynting vector is given by
representing the energy flux due to the electromagnetic field of the particle in motion, but some of the energy stays with the particle, so this is just partly emitted as radiation.
Considering a sphere of radius
centered on the particle in the retarded time, the radiation reaches its surface an instant
after being emitted at
. As the area of a sphere grows proportionally to
, only the terms of the radiation field become relevant, as noted. In this case, the radiation field is perpendicular to
, so the second term in the Poynting vector is zero and we have
The radiation power passing through an element of the surface of the sphere on an instant
is, in terms of a solid angle,
This power is not the same as that produced by the charge at
; rather it is
so the radiated power per unit of solid angle produced by the charge is
The goal of this Demonstration is to find the behavior of the radiation emitted by a point charge with colinear acceleration and velocity. In this case we have
The Poynting vector is thereby
as the angle between
, the acceleration could be parallel or antiparallel so the angle between
, but we have
so this does not modify the result
Finally we have the radiated power per unit of solid angle
In this Demonstration, assume
From this equation we see that as
increases, the radiation is emitted preferentially toward the direction of the particle motion, even if the particle is being decelerated, with
negative. For example, when a high-energy electron is deflected by an atomic nucleus, it emits radiation known as Bremsstrahlung, which can be described by the equations in this Demonstration.
 K. D. Machado, Teoria do Eletromagnetismo
, Vol. 3, Ponta Grossa, Brazil: UEPG, 2006.
 J. D. Jackson, Classical Electrodynamics
, 3rd ed., New York: Wiley, 1999.
 D. J. Griffiths, Introduction to Electrodynamics
, 4th ed., Boston: Pearson, 2013.