Details

Start with the Liénard–Wiechert fields 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

,

where represents the energy flux due to the electromagnetic field of the particle in motion; some of the energy stays with the particle, so this is just partly emitted as radiation.

Considering a sphere of radius centered at 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

,

where

.

The radiation power passing trough 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

or

,

since

.

In this Demonstration we determine the radiation emitted by a point charge in circular motion with constant speed, such that and are perpendicular with constant magnitudes. Considering a counterclockwise circular trajectory of radius in the - plane and center at , when the particle is at 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 in 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 simplification. In this case,

or in Cartesian coordinates:

.

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 when compared with a particle in linear motion, so circular particle accelerators are more efficient when producing radiation than linear ones. These results also show roughly how particle colliders work, such as the Large Hadron Collider (LHC).

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.