Retarded potential
Encyclopedia
The retarded potential formulae describe the scalar or vector potential
for electromagnetic fields of a time-varying
current or charge distribution. The retardation of the influence connecting cause and effect is thereby essential; e.g. the signal takes a finite time, corresponding to the velocity of light, to propagate from the source point
of the field to the point 
where an effect is produced or measured. Otherwise, the formulas below correspond to the same superposition principle acting, e.g., in electrostatics for Coulomb's law
.
These are the electromagnetic
retarded Lorenz gauge potentials for an arbitrary source in free space (vacuum). They satisfy the inhomogeneous
wave equation
s for
and 
in the Lorenz gauge.
Here,
is location, 
is time, and 
is the speed of light
in a vacuum.
is the "retarded time"; the time at which light must be emitted from location 
in order to reach location 
at time t. 
is the electric charge density
, and
is the current density
.
is the dielectric constant
of free space, and
is the magnetic permeability of free space. 
is the electrical potential, and 
is the vector potential
. Finally,
is the integration measure corresponding to 
From
and 
the electromagnetic fields 
and 
can be calculated, 
The corresponding advanced potentials, essentially mathematical objects, are:
The subscript a stands for advanced, and
is the "advanced time". A many-body theory which includes an average of retarded and advanced Liénard-Wiechert potentials
is the Wheeler-Feynman absorber theory also known as the Wheeler-Feynman time-symmetric theory.
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
for electromagnetic fields of a time-varying
Time-variant system
A time-variant system is a system that is not time invariant . Roughly speaking, characteristics of its output depend explicitly upon time.- Overview :...
current or charge distribution. The retardation of the influence connecting cause and effect is thereby essential; e.g. the signal takes a finite time, corresponding to the velocity of light, to propagate from the source point
of the field to the point where an effect is produced or measured. Otherwise, the formulas below correspond to the same superposition principle acting, e.g., in electrostatics for Coulomb's lawCoulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
.
These are the electromagnetic
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
retarded Lorenz gauge potentials for an arbitrary source in free space (vacuum). They satisfy the inhomogeneous
Homogeneous differential equation
The term homogeneous differential equation has several distinct meanings.One meaning is that a first-order ordinary differential equation is homogeneous if it has the formwhere F is a homogeneous function of degree zero; that is to say, that F = F.In a related, but distinct, usage, the term linear...
wave equation
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
s for
and in the Lorenz gauge.Here,
is location, is time, and is the speed of lightSpeed 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...
in a vacuum.
is the "retarded time"; the time at which light must be emitted from location in order to reach location at time t. is the electric charge densityCharge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
, and
is the current densityCurrent density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
.
is the dielectric constantDielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
of free space, and
is the magnetic permeability of free space. is the electrical potential, and is the vector potentialMagnetic potential
The term magnetic potential can be used for either of two quantities in classical electromagnetism: the magnetic vector potential, A, and the magnetic scalar potential, ψ...
. Finally,
is the integration measure corresponding to From
and the electromagnetic fields and can be calculated, The corresponding advanced potentials, essentially mathematical objects, are:
The subscript a stands for advanced, and
is the "advanced time". A many-body theory which includes an average of retarded and advanced Liénard-Wiechert potentialsLiénard-Wiechert Potentials
Liénard-Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential...
is the Wheeler-Feynman absorber theory also known as the Wheeler-Feynman time-symmetric theory.