In
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, the retarded potentials are the
electromagnetic potentials for the
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
generated by
time-varying electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
or
charge distribution
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
s in the past. The fields propagate at the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
''c'', so the delay of the fields connecting
cause and effect
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.
In the Lorenz gauge
The starting point is
Maxwell's equations in the potential formulation using the
Lorenz gauge
In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
:
:
where φ(r, ''t'') is the
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and A(r, ''t'') is the
magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
, for an arbitrary source of
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
ρ(r, ''t'') and
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...
J(r, ''t''), and
is the
D'Alembert operator. Solving these gives the retarded potentials below (all in
SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
).
For time-dependent fields
For time-dependent fields, the retarded potentials are:
:
:
where r is a
point in space, ''t'' is time,
:
is the
retarded time, and d
3r' is the
integration measure using r'.
From φ(r, t) and A(r, ''t''), the fields E(r, ''t'') and B(r, ''t'') can be calculated using the definitions of the potentials:
:
and this leads to
Jefimenko's equations. The corresponding advanced potentials have an identical form, except the advanced time
:
replaces the retarded time.
In comparison with static potentials for time-independent fields
In the case the fields are time-independent (
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for ...
and
magnetostatic
Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equati ...
fields), the time derivatives in the
operators of the fields are zero, and Maxwell's equations reduce to
:
where ∇
2 is the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, which take the form of
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
in four components (one for φ and three for A), and the solutions are:
:
:
These also follow directly from the retarded potentials.
In the Coulomb gauge
In the
Coulomb gauge
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
, Maxwell's equations are
:
:
although the solutions contrast the above, since A is a retarded potential yet φ changes ''instantly'', given by:
:
:
This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but A is not so easily calculable from the current distribution j. However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:
:
:
In linearized gravity
The retarded potential in
linearized general relativity is closely analogous to the electromagnetic case. The trace-reversed tensor
plays the role of the four-vector potential, the
harmonic gauge replaces the electromagnetic Lorenz gauge, the field equations are
, and the retarded-wave solution is
Using SI units, the expression must be divided by
, as can be confirmed by dimensional analysis.
Occurrence and application
A many-body theory which includes an average of retarded and ''advanced''
Liénard–Wiechert potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these desc ...
s is the
Wheeler–Feynman absorber theory
The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assu ...
also known as the Wheeler–Feynman time-symmetric theory.
Example
The potential of charge with uniform speed on a straight line has
inversion in a point
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is inv ...
that is in the recent position. The potential is not changed in the direction of movement.
Feynman, Lecture 26, Lorentz Transformations of the Fields
/ref>
See also
* Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
* Liénard–Wiechert potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these desc ...
* Lenz's law
Lenz's law states that the direction of the electric current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is named after p ...
References
{{Reflist
Potentials