# Radiation Pattern of a Point Charge in Uniform Circular Motion

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A charged particle produces an electric field and, when in motion, also a magnetic field. Furthermore, if accelerated, the particle emits electromagnetic radiation at the speed of the light .

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Contributed by: Deyvid W. M. Pastana, Manuel E. Rodrigues and Luciano J. B. Quaresma (December 2020)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The Liénard–Wiechert potentials for a point particle with charge in arbitrary motion are given by:

and

,

where

,

,

,

,

,

and

.

The electric field has two terms, one proportional to the velocity and the other to the acceleration , varying as and , respectively. Because the acceleration terms dominate at large distances, 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

,

or

,

since

.

The goal of this Demonstration is to find the behavior of the radiation emitted by a point charge in uniform circular motion, when and are perpendicular and have constant magnitudes. Considering a counterclockwise circular trajectory of radius on the - plane and center at , in the exact instant that the particle is on the system origin, the velocity is

;

thus

,

the acceleration is

,

and the radiation is observed on , with distance and direction

,

where is the angle between this direction and the axis, and is the angle between the axis and the projection of on the - plane, as used in spherical coordinates. With these considerations, it is easier to calculate the terms of the radiated power per unit of solid angle:

;

;

;

;

.

These equations lead to

.

In this Demonstration, assume, for simplicity, and consider only the plane of motion. In this case,

.

These results show that as increases, this radiation (usually referred to as synchrotron radiation) is emitted in the direction of the particle motion. The total power has higher intensity than a particle in linear motion, so circular particle accelerators produce radiation more efficiently than linear ones. These results also show roughly how a particle collider (such as the Large Hadron Collider) works.

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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