Liénard–Wiechert potential

The Liénard -Wiechert potentials ( Emil Wiechert introduced independently by Alfred -Marie Liénard (1898 ), named after two Liénard -Wiechert potentials of a moving charge in an essay in 1900 ) describe the electric and magnetic fields that are moved by an electric point charge are generated. They generalize the Coulomb potential, which is generated by a stationary point charge and no magnetic portion has and provide an approximation to represent the potential which would be set by the Doppler effect at high energies.

The scalar Liénard -Wiechert potential is a modified Coulomb potential. The vector potential of which contains the information about the magnetic field is substantially the scalar potential multiplied by the particle velocity.

Compared to the Coulomb potential with the following differences:

• The fields that can be observed at the time are generated from the particles to a previous ( retarded ) timing. The difference is equal to the term from the particle to the observer with the speed of light.
• There is a gain if the particle to the observer moved ( attenuation factor when it moves away ). The gain goes to infinity when the particle velocity to the speed of light goes.

From the potentials of the electric and magnetic field strength can be obtained by derivatives with respect to space and time coordinates (also see potential and wave equation of electrodynamics ). The field strengths fall into a velocity and an acceleration component. The part which contains only the particle velocity is near the particle strongly, at a great distance, however, weak (no far-field ). The acceleration is proportional to the component leads to the radiation of energy to infinity.

The formulas

The position of the particle is considered as a predetermined function. As the trajectory is concluded (eg by electromagnetic fields that exert forces on the particle ) is not considered. In the International System of Units, the Liénard -Wiechert potentials are ( after, however, formulated for fields in matter-free space )

The point on a vector symbol means derivative with respect to time. The subscript " ret" means that particle position and velocity are to be taken at the retarded time. The vector is the unit vector pointing from the sending to the receiving point. For the retarded time, the implicit equation

Apart from the special case of uniform motion, the resolution is roughly approximated by only.

Applications

Synchrotron

Here, the particle moves in a circular orbit at a velocity close to the speed of light. The rate- dependent factor then takes in each cycle at a high peak value. Because if the tangential direction of the velocity coincides with the direction of the observer, that is, when is parallel to, the following applies with

Where the Lorentz factor respectively. The potentials and field strengths are thus proportional. Because the field strengths received quadratic in the radiation energy (see Poynting vector) is the energy of the synchrotron radiation increases proportionately.

Accelerated particles with low velocity at a great distance

Low speed you have, for example, at the beginning of the acceleration process. Large distances of the region which is relevant to electromagnetic radiation. With this specialization, the expressions for the electric and magnetic field strength simplify (see in the limit ). For the magnetic field

The electric field strength follows from a general relation for fields in the far zone

Thus, the almost infinite energy flux density ( Poynting vector) is at a distance equal in magnitude

Wherein the angle between the acceleration vector and the observation direction. The energy flux per unit solid angle is obtained by omitting the denominator.

Derivation of the B field: The particle should be small compared to the speed of light, so that all terms may be ignored, which contain a factor in the outcome. If derivatives act according to the retarded time, so does the particle location not to be mitdifferenziert. Thus, to a first approximation

When derivatives of the vector potential, the factor can not stand still; So anyway, only this factor needs to be differentiated. For the magnetic field, one obtains

With a chain rule was used for the rotation. It has also been used.