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In electrodynamics, the **retarded potentials** are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light *c*, so the delay of the fields connecting cause and effect 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.^{[1]}

The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge:

where φ(**r**, *t*) is the electric potential and **A**(**r**, *t*) is the magnetic vector potential, for an arbitrary source of charge density ρ(**r**, *t*) and current density **J**(**r**, *t*), and is the D'Alembert operator.^{[2]} Solving these gives the retarded potentials below (all in SI units).

For time-dependent fields, the retarded potentials are:^{[3]}^{[4]}

where **r** is a point in space, *t* is time,

is the retarded time, and d^{3}**r'** 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 the case the fields are time-independent (electrostatic and magnetostatic fields), the time derivatives in the operators of the fields are zero, and Maxwell's equations reduce to

where ∇^{2} is the Laplacian, which take the form of Poisson's equation 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, Maxwell's equations are^{[5]}

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:

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^{[6]}

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.

The potential of charge with uniform speed on a straight line has inversion in a point that is in the recent position. The potential is not changed in the direction of movement.^{[7]}

**^**Rohrlich, F (1993). "Potentials". In Parker, S.P. (ed.).*McGraw Hill Encyclopaedia of Physics*(2nd ed.). New York. p. 1072. ISBN 0-07-051400-3.`{{cite encyclopedia}}`

: CS1 maint: location missing publisher (link)**^**Garg, A.,*Classical Electromagnetism in a Nutshell*, 2012, p. 129**^**Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9**^**Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3**^**Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3**^**Sean M. Carroll, "Lecture Notes on General Relativity" (arXiv:gr-qc/9712019), equations 6.20, 6.21, 6.22, 6.74**^**Feynman, Lecture 26, Lorentz Transformations of the Fields