# 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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This Demonstration shows the angular distribution of the power of the electromagnetic radiation due to a point charge in uniform circular motion using Lorentz gauge in Cartesian coordinates. This is a function of the velocity through the parameter . As approaches , the radiation points toward the direction of motion with higher intensity compared to a particle in linear motion. Because of this, circular particle accelerators, such as synchrotrons and betatrons, produce radiation more efficiently.

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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Contributed by: Deyvid W. M. Pastana, Manuel E. Rodrigues and Luciano J. B. Quaresma (December 2020)
Open content licensed under CC BY-NC-SA

## 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:

;

;

;

;

.

.

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.

## Permanent Citation

Deyvid W. M. Pastana, Manuel E. Rodrigues and Luciano J. B. Quaresma

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