Start 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 the other by the acceleration

, proportional to

and

, respectively. Because at large distances the acceleration terms dominate, these are also known as radiation fields. The Poynting vector is given by

,

which represents the energy flux due to the electromagnetic field of the particle in motion. Some of the energy stays with the particle, while the remainder represents emitted radiation.

Consider 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 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,

and

.

The Poynting vector is therefore

and defining

as the angle between

and

, from

, the acceleration could be parallel or antiparallel so the angle between

and

could be

or

, but

, so this does not modify the result

.

Finally, the radiated power per unit of solid angle is

,

.

In this Demonstration, assume

for simplification.

This equation implies 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.

[1] K. D. Machado,

*Teoria do Eletromagnetismo*, Vol. 3, Ponta Grossa, Brazil: UEPG, 2006.

[2] J. D. Jackson,

*Classical Electrodynamics*, 3rd ed., New York: Wiley, 1999.

[3] D. J. Griffiths,

*Introduction to Electrodynamics*, 4th ed., Boston: Pearson, 2013.