Liénard-Wiechert Potentials

Liénard-Wiechert Potentials

Discussion

Encyclopedia
Liénard-Wiechert potentials describe the classical electromagnetic
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

effect of a moving electric point charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

in terms of a vector potential
Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....

and a scalar potential
Scalar potential
A scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...

. Built directly from Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

, these potentials describe the complete, relativistically
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

correct, time-varying electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

for a point charge in arbitrary motion, but are not corrected for quantum-mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

in the form of waves can be obtained from these potentials.

These expressions were developed in part by Alfred-Marie Liénard
Alfred-Marie Liénard
Alfred-Marie Liénard , was a French physicist and engineer. He is most well known for his invention of the Liénard–Wiechert potentials....

in 1898 and independently by Emil Wiechert
Emil Wiechert
Emil Johann Wiechert was a German geophysicist who presented the first verifiable model of a layered structure of the Earth.-Life:...

in 1900 and continued into the early 1900s.

The Liénard-Wiechert potentials can be generalized according to gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

.

The Liénard-Wiechert potentials are the initial terms in an expansion of retarded potential solutions of the nonhomogeneous wave equations
(the retarded Lorentz-gauge potentials) in terms of co-moving moments of localized, time-dependent, moving charges and currents; and the following terms give explicit expressions for retarded potential solutions related to moving dipoles and quadrupoles.

Implications

The study of classical electrodynamics was instrumental in Einstein's development of the theory of relativity. Analysis of the motion and propagation of electromagnetic waves led to the special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

description of space and time. The Liénard–Wiechert formulation is an important launchpad into more complex analysis of relativistic moving particles.

The Liénard–Wiechert description is accurate for a large, independent moving particle, but breaks down at the quantum level.

Quantum mechanics sets important constraints on the ability of a particle to emit radiation. The classical formulation, as laboriously described by these equations, expressly violates experimentally observed phenomena. For example, an electron around an atom
Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

does not emit radiation in the pattern predicted by these classical equations. Instead, it is governed by quantized principles regarding its energy state. In the later decades of the twentieth century, quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

helped bring together the radiative behavior with the quantum constraints.

Universal Speed Limit

The force on a particle at a given location and time depends in a complicated way on the position of the source particles at an earlier time due to the finite speed, c
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

, at which electromagnetic information travels. A particle on Earth 'sees' a charged particle accelerate on the Moon as this acceleration happened 1.5 seconds ago, and a charged particle's acceleration on the Sun as happened 500 seconds ago. This earlier time in which an event happens such that a particle at location 'sees' this event at a later time is called the retarded time, . The retarded time varies with position; for example the retarded time at the Moon is 1.5 seconds before the current time and the retarded time on the Sun is 500 s before the current time. The retarded time can be calculated as:

where is the distance of the particle from the source at the retarded time. Only electromagnetic wave effects depends fully on the retarded time.

A novel feature in the Liénard–Wiechert potential is seen in the breakup of its terms into two types of field terms (see below), only one of which depends fully on the retarded time. The first of these is the static electric field term, and depends only on the distance to the moving charge; the other term is dynamic in that it requires that the moving charge be accelerating with a component perpendicular to the line connecting the charge and the observer. This second term is connected with electromagnetic radiation.

The first term describes near field
Near and far field
The near field and far field and the transition zone are regions of the electromagnetic radiation field that emanates from a transmitting antenna, or as a result of radiation scattering off an object...

effects from the charge, and its direction in space is updated with a term that corrects for any constant-velocity motion of the charge on its distant static field, so that the distant static field appears at distance from the charge, with no aberration of light
Aberration of light
The aberration of light is an astronomical phenomenon which produces an apparent motion of celestial objects about their real locations...

or light-time correction
Light-time correction
Light-time correction is a displacement in the apparent position of a celestial object from its true position caused by the object's motion during the time it takes its light to reach an observer....

. This term, which corrects for time-retardation delays in the direction of the static field, is required by Lorentz invariance. A charge moving with a constant velocity must appear to a distant observer in exactly the same way as a static charge appears to a moving observer, and in the latter case, the direction of the static field must change instantaneously, with no time-delay. Thus, static fields (the first term) point exactly at the true position of the object, if its velocity has not changed over the retarded time delay.

The second term, however, which contains information about the acceleration and other unique behavior of the charge that cannot be removed by changing the Lorentz frame (inertial reference frame of the observer), is fully dependent for direction on the time-retarded position of the source. Thus, electromagnetic radiation (described by the second term) always appears to come from the direction to the position of the emitting charge at the retarded time. Only this second term describes information transfer about the behavior of the charge, which transfer occurs (radiates from the charge) at the speed of light. At "far" distances (longer than several wavelengths of radiation), the 1/R dependence of this term makes electromagnetic field effects (the value of this field term) more powerful than "static" field effects, which are described by the 1/R2 potential of the first (static) term and thus decay more rapidly with distance from the charge.

Definition of Liénard-Wiechert potentials

The Liénard-Wiechert potentials (scalar potential field) and (vector potential field) are for a source point charge at position traveling with velocity :

and

where .

Corresponding values of electric and magnetic fields

We can calculate the electric and magnetic fields directly from the potentials using the definitions:
and

The calculation is non trivial and requires a number of steps. The electric and magnetic fields are (in non-covariant form):

and

where , and (the Lorentz factor
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...

).

Note that the part of the first term updates the direction of the field toward the instantantaneous position of the charge, if it continues to move with constant velocity . The second term, which is connected with electromagnetic radiation by the moving charge, requires charge acceleration and if this is zero, the value of this term is zero, and the charge does not radiate. This term requires additionally that a component of the charge acceleration be in a direction transverse to the line which connects the charge and the observer of the field . The direction of the field associated with this radiative term is toward the fully time-retarded position of the charge (i.e. where the charge was when it was accelerated).

Retarded potential solutions

In the case that there are no boundaries surrounding the sources, the retarded solutions for the scalar and vector potentials (CGS units) of the nonhomogeneous wave equations with sources given by the charge and current densities and are (see Nonhomogeneous electromagnetic wave equation)

and

where

is a Dirac delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

. For a moving point charge at traveling with velocity , the current and charge densities are

and the retarded potential solutions simplify to the Liénard-Wiechert potentials.

• Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

which govern classical electromagnetism
• Classical electromagnetism
Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

for the larger theory surrounding this analysis
• Relativistic electromagnetism
Relativistic electromagnetism
Relativistic electromagnetism is a modern teaching strategy for developing electromagnetic field theory from Coulomb’s law and Lorentz transformations. Though Coulomb’s law expresses action at a distance, it is an easily understood electric force principle...

• Special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

, which was a direct consequence of these analyses
• Rydberg formula
Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on November 5, 1888.-History:...

for quantum description of the EM radiation due to atomic orbital electrons
• Jefimenko's equations
Jefimenko's equations
In electromagnetism, Jefimenko's equations describe the behavior of the electric and magnetic fields in terms of the charge and current distributions at retarded times....

• Larmor formula
Larmor formula
In physics, in the area of electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J...

• Abraham-Lorentz force
Abraham-Lorentz force
In the physics of electromagnetism, the Abraham–Lorentz force is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. It is also called the radiation reaction force....

• Inhomogeneous electromagnetic wave equation
Inhomogeneous electromagnetic wave equation
Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a inhomogeneous electromagnetic wave equation with sources...

• Wheeler-Feynman absorber theory also known as the Wheeler-Feynman time-symmetric theory