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| {{electromagnetism}}
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| In [[electrodynamics]], the '''retarded potentials''' are the [[electromagnetic potential]]s for the [[electromagnetic field]] generated by [[Time-variant system|time-varying]] [[electric current]] or [[charge distribution]]s in the past. The fields propagate at the [[speed of light]] ''c'', so the delay of the fields connecting [[Causality (physics)|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.<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3</ref>
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| ==Potentials in the Lorenz gauge==
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| {{main|Maxwell's equations|Mathematical descriptions of the electromagnetic field}}
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| [[File:Universal charge distribution.svg|250px|right|thumb|Position vectors '''r''' and '''r′''' used in the calculation.]]
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| The starting point is [[Maths of EM field#Potential field approach|Maxwell's equations in the potential formulation]] using the [[Lorenz gauge]]:
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| :<math> \Box \varphi = - \dfrac{\rho}{\epsilon_0} \,,\quad \Box \mathbf{A} = -\mu_0\mathbf{J}</math>
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| 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 <math>\Box</math> is the [[D'Alembert operator]]. Solving these gives the retarded potentials below. | |
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| ===Non-Retarded and advanced potentials for time-dependent fields===
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| For time-dependent fields, the retarded potentials are:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 9-780471-927129</ref><ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
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| :<math> \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>
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| :<math>\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>
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| where '''r''' is a [[position vector|point]] in space, ''t'' is time,
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| :<math>t_r = t-\frac{|\mathbf r - \mathbf r'|}{c}</math> | |
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| is the [[retarded time]], and d<sup>3</sup>'''r'''' is the [[List of integration and measure theory topics|integration measure]] using '''r''''.
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| 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:
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| :<math>-\mathbf{E} = \nabla\varphi +\frac{\partial\mathbf{A}}{\partial t}\,,\quad \mathbf{B}=\nabla\times\mathbf A\,.</math>
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| and this leads to [[Jefimenko's equations]]. The corresponding advanced potentials have an identical form, except the advanced time
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| :<math>t_a = t+\frac{|\mathbf r - \mathbf r'|}{c}</math>
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| replaces the retarded time.
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| ===Comparison with static potentials for time-independent fields===
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| In the case the fields are time-independent ([[electrostatic]] and [[magnetostatic]] fields), the time derivatives in the <math>\Box</math> operators of the fields are zero, and Maxwell's equations reduce to
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| :<math> \nabla^2 \varphi =-\dfrac{\rho}{\epsilon_0}\,,\quad \nabla^2 \mathbf{A} =- \mu_0 \mathbf{J}\,,</math> | |
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| where ∇<sup>2</sup> 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:
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| :<math> \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>
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| :<math>\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>
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| These also follow directly from the retarded potentials.
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| ==Potentials in the Coulomb gauge==
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| In the [[Coulomb gauge]], Maxwell's equations are<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
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| :<math> \nabla^2 \varphi =-\dfrac{\rho}{\epsilon_0}</math>
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| :<math> \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>
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| although the solutions contrast the above, since '''A''' is a retarded potential yet φ changes ''instantly'', given by:
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| :<math>\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>
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| :<math> \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>
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| 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:
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| :<math>\varphi(\mathbf{r}, t) = \dfrac{1}{4\pi}\int \dfrac{\nabla \cdot \mathbf{E}({r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}'</math>
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| :<math> \mathbf{A}(\mathbf{r},t) = \dfrac{1}{4\pi}\int \dfrac{\nabla \times \mathbf{B}({r}',t)}{|\mathbf r - \mathbf r'|}\mathrm{d}^3\mathbf{r}'</math>
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| ==Occurrence and application==
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| 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.
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| ==References==
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| {{Reflist}}
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| [[Category:Potentials]]
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Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. I've always loved residing in Alaska. Credit authorising is exactly where my primary earnings comes from. To perform domino is something I really enjoy performing.
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