# Physics:Retarded potential

__: Type of potential in electrodynamics__

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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]}

## In the Lorenz gauge

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

- [math]\displaystyle{ \Box \varphi = \dfrac{\rho}{\epsilon_0} \,,\quad \Box \mathbf{A} = \mu_0\mathbf{J} }[/math]

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 [math]\displaystyle{ \Box }[/math] is the D'Alembert operator.^{[2]} Solving these gives the retarded potentials below (all in SI units).

### For time-dependent fields

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

- [math]\displaystyle{ \mathrm\varphi (\mathbf r , t) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho (\mathbf r' , t_r)}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r' }[/math]

- [math]\displaystyle{ \mathbf A (\mathbf r , t) = \frac{\mu_0}{4\pi}\int \frac{\mathbf J (\mathbf r' , t_r)}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r'\,. }[/math]

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

- [math]\displaystyle{ t_r = t-\frac{|\mathbf r - \mathbf r'|}{c} }[/math]

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:

- [math]\displaystyle{ -\mathbf{E} = \nabla\varphi +\frac{\partial\mathbf{A}}{\partial t}\,,\quad \mathbf{B}=\nabla\times\mathbf A\,. }[/math]

and this leads to Jefimenko's equations. The corresponding advanced potentials have an identical form, except the advanced time

- [math]\displaystyle{ t_a = t+\frac{|\mathbf r - \mathbf r'|}{c} }[/math]

replaces the retarded time.

### In 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 [math]\displaystyle{ \Box }[/math] operators of the fields are zero, and Maxwell's equations reduce to

- [math]\displaystyle{ \nabla^2 \varphi =-\dfrac{\rho}{\epsilon_0}\,,\quad \nabla^2 \mathbf{A} =- \mu_0 \mathbf{J}\,, }[/math]

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:

- [math]\displaystyle{ \mathrm\varphi (\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho (\mathbf r' )}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r' }[/math]

- [math]\displaystyle{ \mathbf A (\mathbf{r}) = \frac{\mu_0}{4\pi}\int \frac{\mathbf J (\mathbf r' )}{|\mathbf r - \mathbf r'|}\, \mathrm{d}^3\mathbf r'\,. }[/math]

These also follow directly from the retarded potentials.

## In the Coulomb gauge

In the Coulomb gauge, Maxwell's equations are^{[5]}

- [math]\displaystyle{ \nabla^2 \varphi =-\dfrac{\rho}{\epsilon_0} }[/math]

- [math]\displaystyle{ \nabla^2 \mathbf{A} - \dfrac{1}{c^2}\dfrac{\partial^2 \mathbf{A}}{\partial t^2}=- \mu_0 \mathbf{J} +\dfrac{1}{c^2}\nabla\left(\dfrac{\partial \varphi}{\partial t}\right)\,, }[/math]

although the solutions contrast the above, since **A** is a retarded potential yet φ changes *instantly*, given by:

- [math]\displaystyle{ \varphi(\mathbf{r}, t) = \dfrac{1}{4\pi\epsilon_0}\int \dfrac{\rho(\mathbf{r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}' }[/math]

- [math]\displaystyle{ \mathbf{A}(\mathbf{r},t) = \dfrac{1}{4\pi \varepsilon_0} \nabla\times\int \mathrm{d}^3\mathbf{r'} \int_0^{|\mathbf{r}-\mathbf{r}'|/c} \mathrm{d}t_r \dfrac{ t_r \mathbf{J}(\mathbf{r'}, t-t_r)}{|\mathbf{r}-\mathbf{r}'|^3}\times (\mathbf{r}-\mathbf{r}') \,. }[/math]

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:

- [math]\displaystyle{ \varphi(\mathbf{r}, t) = \dfrac{1}{4\pi}\int \dfrac{\nabla \cdot \mathbf{E}(\mathbf{r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}' }[/math]

- [math]\displaystyle{ \mathbf{A}(\mathbf{r},t) = \dfrac{1}{4\pi}\int \dfrac{\nabla \times \mathbf{B}(\mathbf{r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}' }[/math]

## In linearized gravity

The retarded potential in linearized general relativity is closely analogous to the electromagnetic case. The trace-reversed tensor [math]\displaystyle{ \tilde h_{\mu\nu} = h_{\mu\nu} - \frac 1 2 \eta_{\mu\nu} h }[/math] plays the role of the four-vector potential, the harmonic gauge [math]\displaystyle{ \tilde h^{\mu\nu}{}_{,\mu} = 0 }[/math] replaces the electromagnetic Lorenz gauge, the field equations are [math]\displaystyle{ \Box \tilde h_{\mu\nu} = -16\pi G T_{\mu\nu} }[/math], and the retarded-wave solution is^{[6]}
[math]\displaystyle{ \tilde h_{\mu\nu}(\mathbf r, t) = 4 G \int \frac{T_{\mu\nu}(\mathbf r', t_r)}{|\mathbf r - \mathbf r'|} \mathrm d^3 \mathbf r'. }[/math]
Using SI units, the expression must be divided by [math]\displaystyle{ c^4 }[/math], as can be confirmed by dimensional analysis.

## 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.

## Example

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]}

## See also

## References

- ↑ Rohrlich, F (1993). "Potentials". in Parker, S.P..
*McGraw Hill Encyclopaedia of Physics*(2nd ed.). New York. p. 1072. ISBN 0-07-051400-3. https://archive.org/details/mcgrawhillencycl1993park/page/1072/. - ↑ 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

Original source: https://en.wikipedia.org/wiki/Retarded potential.
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