# Retarded potential

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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]}

## Potentials in the Lorenz gauge

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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 potential, for an arbitrary source of charge density ρ(**r**, *t*) and current density **J**(**r**, *t*), and is the D'Alembert operator. Solving these gives the retarded potentials below.

### Retarded and advanced potentials for time-dependent fields

For time-dependent fields, the retarded potentials are:^{[2]}^{[3]}

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.

### Comparison with static potentials for time-independent fields

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.

## Potentials in the Coulomb gauge

In the Coulomb gauge, Maxwell's equations are^{[4]}

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:

## Occurrence and application

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.

## References

- ↑ McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
- ↑ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 9-780471-927129
- ↑ 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