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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.
 
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]]

Revision as of 17:33, 28 January 2014

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As measured in a given frame of reference, and for a given 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 covariant.

Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge.

In this article, index notation and the Minkowski 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 SI units and Gaussian-cgs units.

Definition

The electromagnetic four-potential can be defined as:[2]

SI units cgs units
Aα=(ϕ/c,A) Aα=(ϕ,A)

in which ϕ is the electric potential, and A is the magnetic potential (a vector potential). The units of Aα are V·s·m−1 in SI, and Mx·cm−1 in Gaussian-cgs.

The electric and magnetic fields associated with these four-potentials are:[3]

SI units cgs units
E=ϕAt E=ϕ1cAt
B=×A. B=×A.

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:

Fμν=μAννAμ.

This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition.

In the Lorenz gauge

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Often, the Lorenz gauge condition αAα=0 in an inertial frame of reference is employed to simplify Maxwell's equations as:[4]

SI units cgs units
Aα=μ0Jα Aα=4πcJα

where Jα are the components of the four-current, and

=1c22t22

is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes:

SI units cgs units
ϕ=ρϵ0 ϕ=4πρ
A=μ0j A=4πcj

For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are:[5]

ϕ(r,t)=14πϵ0d3xρ(r,tr)|rr|
A(r,t)=μ04πd3xj(r,tr)|rr|,

where

tr=t|rr|c

is the retarded time. This is sometimes also expressed with

ρ(r,tr)=[ρ(r,t)],

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 inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. 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 −2 (the induction field) and a component decreasing as r −1 (the radiation field).

See also

References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  1. Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
  2. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
  3. Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
  4. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
  5. Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9