Angular Distribution of Radiated Power Emitted by an Accelerated Point Charge

A charged particle naturally produces an electric field and, when in motion, also a magnetic field. Furthermore, if accelerated, the particle also emits electromagnetic radiation at the speed of light .
This Demonstration shows the angular distribution of power of the electromagnetic radiation due to a point charge with collinear velocity and acceleration using Lorentz gauge in Cartesian coordinates. This is a function of the velocity through the parameter . As the speed approaches the speed of light, the radiation points toward the direction of motion whether the particle is being accelerated or decelerated. The graphic represents the angular distribution of power in the moving frame of the particle. For simplicity, the acceleration is assumed to be constant, such that .

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We begin with the Liénard–Wiechert potentials for a point particle with charge in arbitrary motion:
and
,
where
,
and
.
The electric field has two terms, one determined by the velocity and other by the acceleration , respectively proportional to and . 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 and
.
The Poynting vector is thereby
and defining as the angle between and , from , the acceleration could be parallel or antiparallel so the angle between and could be or , but we have so this does not modify the result
.
Finally we have the radiated power per unit of solid angle
,
which gives us
.
In Cartesian coordinates
.
In this Demonstration, assume for simplification.
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.
References
[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.
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