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{{electromagnetism|cTopic=[[Covariant formulation of classical electromagnetism|Covariant formulation]]}}
An '''electromagnetic four-potential''' is a [[General relativity|relativistic]] [[Euclidean vector|vector]] function from which the [[electromagnetic field]] can be derived. It combines both an [[electric scalar potential]] and a [[magnetic vector potential]] into a single [[four-vector]]. <ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>


As measured in a given [[frame of reference]], and for a given [[Gauge theory|gauge]], the first component of the electromagnetic four-potential is the electric scalar potential, and the other three components make up the magnetic vector potential. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is [[Lorentz covariance|Lorentz covariant]].


Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge.
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In this article, [[index notation]] and the [[Minkowski metric|Minkowski]] [[metric signature|metric]] (+−−−) will be used, see also [[Ricci calculus]], [[covariance and contravariance of vectors]] and [[raising and lowering indices]] for more details on notation. Formulas are given in [[International System of Units|SI units]] and [[Gaussian units|Gaussian-cgs units]].
 
== Definition ==
 
The '''electromagnetic four-potential''' can be defined as:<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
 
:{| class="wikitable"
|-
! SI units
! cgs units
|-
||<math>A^\alpha = \left( \phi / c , \mathbf{A} \right)\,\!</math> || <math>A^\alpha = (\phi, \mathbf{A})</math>
|-
|}
 
in which ''ϕ'' is the [[electric potential]], and '''A''' is the [[magnetic potential]] (a [[vector potential]]). The units of ''A<sup>α</sup>'' are [[volt|V]]·[[second|s]]·[[metre|m]]<sup>−1</sup> in SI, and [[maxwell (unit)|Mx]]·[[centimeter|cm]]<sup>−1</sup> in [[Gaussian units|Gaussian-cgs]].
 
The electric and magnetic fields associated with these four-potentials are:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9</ref>
 
:{| class="wikitable"
|-
! SI units
! cgs units
|-
||<math>\mathbf{E} = -\mathbf{\nabla} \phi - \frac{\partial \mathbf{A}}{\partial t}</math>||<math>\mathbf{E} = -\mathbf{\nabla} \phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} </math>
|-
||<math>\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}. </math>||<math>\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}. </math>
|-
|}
 
In [[special relativity]], the electric and magnetic fields must be written in the form of a [[tensor]] so they transform correctly under [[Lorentz transformations]] - achieved by the [[electromagnetic tensor]]. This is written in terms of the electromagnetic four-potential as:
 
:<math>F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.</math>
 
This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.
 
== In the Lorenz gauge ==
{{main|mathematical descriptions of the electromagnetic field|Retarded potential}}
Often, the [[Lorenz gauge condition]] <math>\partial_{\alpha} A^{\alpha} = 0</math> in an [[inertial frame of reference]] is employed to simplify [[Maxwell's equations]] as:<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3</ref>
 
:{| class="wikitable"
|-
! SI units
! cgs units
|-
||<math>\Box A^\alpha = \mu_0 J^\alpha</math> ||<math> \Box A^\alpha = \frac{4 \pi}{c} J^\alpha </math>
|-
|}
 
where ''J<sup>α</sup>'' are the components of the [[four-current]], and
 
:<math>\Box = \frac{1}{c^2} \frac{\partial^2} {\partial t^2}-\nabla^2</math>
 
is the [[d'Alembertian]] operator. In terms of the scalar and vector potentials, this last equation becomes:
 
:{| class="wikitable"
|-
! SI units
! cgs units
|-
||<math>\Box \phi = \frac{\rho}{\epsilon_0}</math> || <math>\Box \phi = 4 \pi \rho</math>
|-
||<math>\Box \mathbf{A} = \mu_0 \mathbf{j}</math>||<math>\Box \mathbf{A} = \frac{4 \pi}{c} \mathbf{j} </math>
|-
|}
 
For a given charge and current distribution, ρ('''r''', ''t'') and '''j'''('''r''', ''t''), the solutions to these equations in SI units are:<ref>Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9</ref>
 
:<math>\phi (\mathbf{r}, t) = \frac{1}{4 \pi \epsilon_0} \int \mathrm{d}^3 x^\prime \frac{\rho( \mathbf{r}^\prime, t_r)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|}</math>
 
:<math>\mathbf A (\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \mathrm{d}^3 x^\prime \frac{\mathbf{j}( \mathbf{r}^\prime, t_r)}{ \left| \mathbf{r} - \mathbf{r}^\prime \right|},</math>
 
where
:<math>t_r = t - \frac{\left|\mathbf{r}-\mathbf{r}'\right|}{c}</math>
is the [[retarded time]]. This is sometimes also expressed with
:<math>\rho(\mathbf{r}',t_r)=[\rho(\mathbf{r}',t)],</math>  
where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an [[Homogeneous differential equation|inhomogeneous]] [[differential equation]], any solution to the homogeneous equation can be added to these to satisfy the [[boundary condition]]s. These homogeneous solutions in general represent waves propagating from sources outside the boundary.
 
When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying according to ''r''<sup> −2</sup> (the induction field) and a component decreasing as ''r''<sup> −1</sup>  (the radiation field).
 
== See also ==
 
* [[Covariant formulation of classical electromagnetism]]
* [[Jefimenko's equations]]
* [[Gluon field]]
 
== References ==
{{reflist}}
* {{cite book | author = Rindler, Wolfgang | title=Introduction to Special Relativity (2nd)| location= Oxford | publisher=Oxford University Press | year=1991 | isbn=0-19-853952-5}}
* {{cite book | author = Jackson, J D | title=Classical Electrodynamics (3rd) | location =New York | publisher=Wiley | year = 1999 | id=ISBN ISBN 0-471-30932-X }}
 
[[Category:Theory of relativity]]
[[Category:Electromagnetism]]

Latest revision as of 02:54, 12 November 2014


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