Editing Magnetic vector potential

{{Use American English|date=March 2019}}{{Short description|Integral of the magnetic field}}
{{Electromagnetism|Magnetostatics}}

In [[classical electromagnetism]], '''magnetic vector potential''' (often denoted '''A''') is the [[vector quantity]] defined so that its [[Curl (mathematics)|curl]] is equal to the [[magnetic field]], '''B''': <math display="inline"> \nabla \times \mathbf{A} = \mathbf{B}</math>. Together with the [[electric potential]] ''φ'', the magnetic vector potential can be used to specify the [[electric field]] '''E''' as well. Therefore, many equations of electromagnetism can be written either in terms of the fields '''E''' and '''B''', or equivalently in terms of the potentials ''φ'' and '''A'''. In more advanced theories such as [[quantum mechanics]], most equations use potentials rather than fields.

Magnetic vector potential was independently introduced by [[Franz Ernst Neumann]]<ref>{{cite journal |last=Neumann |first=Franz Ernst |date=January 1, 1846 |title=Allgemeine Gesetze der induzirten elektrischen Ströme (General laws of induced electrical currents) |url=https://zenodo.org/record/1423608 |journal=Annalen der Physik |volume=143 |issue=11 |pages=31–34 |doi=10.1002/andp.18461430103}}</ref> and [[Wilhelm Eduard Weber]]<ref>W. E. Weber, Elektrodymische Maassbestimungen, uber ein allgemeines Grundgesetz der elektrischen Wirkung, Abhandlungen bei Begrund der Koniglichen Sachsischen Gesellschaft der Wissenschaften (Leipzig, 1846), pp. 211–378 [W. E. Weber, Wilhelm Weber’s Werkes, Vols. 1–6 (Berlin, 1892–1894); Vol. 3, pp. 25–214]. </ref> in 1845 and in 1846, respectively to discuss [[Ampère's circuital law]].<ref>{{Cite journal |last1=Wu |first1=A. C. T. |last2=Yang |first2=Chen Ning |date=2006-06-30 |title=Evolution of the Concept of the Vector Potential in the Description of Fundamental Interactions |url=https://www.worldscientific.com/doi/abs/10.1142/S0217751X06033143 |journal=International Journal of Modern Physics A |language=en |volume=21 |issue=16 |pages=3235–3277 |doi=10.1142/S0217751X06033143 |bibcode=2006IJMPA..21.3235W |issn=0217-751X|url-access=subscription }}</ref>  [[Lord Kelvin|William Thomson]] also introduced the modern version of the vector potential in 1847, along with the formula relating it to the magnetic field.<ref>{{cite journal | last =Yang | first =ChenNing | title =The conceptual origins of Maxwell's equations and gauge theory | journal =Physics Today | volume =67 | issue =11 | pages =45–51 | date =2014 | doi =10.1063/PT.3.2585 | bibcode =2014PhT....67k..45Y }}</ref>

== Unit conventions ==
This article uses the SI system.

In the [[SI system]], the units of '''A''' are [[volt|V]]·[[second|s]]·[[metre|m]]<sup>−1</sup> or [[Weber_(unit)|Wb]]·[[metre|m]]<sup>−1</sup> and are the same as that of [[momentum]] per unit [[electric charge|charge]], or [[force]] per unit [[Electric current|current]].

== Definition ==
The magnetic vector potential, <math>\mathbf{A}</math>, is a [[vector field]], and the [[electric potential]], <math>\phi</math>, is a [[scalar field]] such that:<ref name=Feynman1515>{{harvp|Feynman|1964|loc=chpt. 15}}</ref>
<math display="block">\mathbf{B} = \nabla \times \mathbf{A}\ , \quad \mathbf{E} = -\nabla \phi - \frac{ \partial \mathbf{A} }{ \partial t },</math>
where <math>\mathbf{B}</math> is the [[magnetic field]] and <math>\mathbf{E} </math> is the [[electric field]]. In [[magnetostatics]] where there is no time-varying current or [[charge distribution]], only the first equation is needed. (In the context of [[electrodynamics]], the terms ''vector potential'' and ''scalar potential'' are used for ''magnetic vector potential'' and ''[[electric potential]]'', respectively. In mathematics, [[vector potential]] and [[scalar potential]] can be generalized to higher dimensions.)

If electric and magnetic fields are defined as above from potentials, they automatically satisfy two of [[Maxwell's equations]]: [[Gauss's law for magnetism]] and [[Faraday's law of induction|Faraday's law]]. For example, if <math>\mathbf{A}</math> is continuous and well-defined everywhere, then it is guaranteed not to result in [[magnetic monopole]]s. (In the mathematical theory of magnetic monopoles, <math>\mathbf{A}</math> is allowed to be either undefined or multiple-valued in some places; see [[magnetic monopole]] for details).

Starting with the above definitions and remembering that the divergence of the curl is zero and the curl of the gradient is the zero vector:
<math display="block">\begin{align}
   \nabla \cdot \mathbf{B} &= \nabla \cdot \left(\nabla \times \mathbf{A}\right) = 0\ ,\\
  \nabla \times \mathbf{E} &= \nabla \times \left( -\nabla\phi - \frac{ \partial\mathbf{A} }{ \partial t } \right) = -\frac{ \partial }{ \partial t } \left(\nabla \times \mathbf{A}\right) = -\frac{ \partial \mathbf{B} }{ \partial t } ~.
\end{align}</math>

Alternatively, the existence of <math>\mathbf{A}</math> and <math>\phi</math> is guaranteed from these two laws using [[Helmholtz decomposition|Helmholtz's theorem]]. For example, since the magnetic field is [[divergence]]-free (Gauss's law for magnetism; i.e., <math>\nabla \cdot \mathbf{B} = 0</math>), <math> \mathbf{A}</math> always exists that satisfies the above definition.

The vector potential <math>\mathbf{A} </math> is used when studying the [[Lagrangian mechanics|Lagrangian]] in [[classical mechanics]] and in [[quantum mechanics]] (see [[Pauli equation|Schrödinger equation for charged particles]], [[Dirac equation]], [[Aharonov–Bohm effect]]).

In [[minimal coupling]], <math>q \mathbf{A} </math> is called the potential momentum, and is part of the [[canonical momentum]].

The [[line integral]] of <math>\mathbf{A}</math> over a closed loop, <math>\Gamma</math>, is equal to the [[magnetic flux]], <math>\Phi_{\mathbf{B}}</math>, through a surface, <math>S</math>, that it encloses:
<math display="block">\oint_\Gamma \mathbf{A}\, \cdot\ d{\mathbf{\Gamma}} = \iint_S \nabla\times\mathbf{A}\ \cdot\ d \mathbf{S} = \Phi_\mathbf{B} ~.</math>

Therefore, the units of <math>\mathbf{A}</math> are also equivalent to [[weber (unit)|weber]] per [[metre]]. The above equation is useful in the [[flux quantization]] of [[Superconductor|superconducting loops]].

In the Coulomb gauge <math> \nabla \cdot \mathbf{A} = 0 </math>, there is a formal analogy between the relationship between the vector potential and the magnetic field to [[Ampere's law]] <math> \nabla \times \mathbf{B} = \mu_0 \mathbf{J} </math>. Thus, when finding the vector potential of a given magnetic field, one can use the same methods one uses when finding the magnetic field given a current distribution.

Although the magnetic field, <math>\mathbf{B}</math>, is a [[pseudovector]] (also called [[axial vector]]), the vector potential, <math>\mathbf{A}</math>, is a [[polar vector]].<ref name=Fitzpatrick>{{cite web |first=Richard |last=Fitzpatrick |title=Tensors and pseudo-tensors |type=lecture notes |publisher=[[University of Texas]] |place=Austin, TX |url=http://farside.ph.utexas.edu/teaching/em/lectures/node120.html }}</ref>  This means that if the [[right-hand rule]] for [[cross product]]s were replaced with a left-hand rule, but without changing any other equations or definitions, then <math>\mathbf{B}</math> would switch signs, but '''A''' would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa.<ref name=Fitzpatrick/>

== Magnetostatics in the Coulomb Gauge ==
In [[magnetostatics]], if the Coulomb gauge <math>\ \nabla \cdot \mathbf{A} = 0 </math> is imposed, then there is an analogy between <math> \mathbf{A}, \mathbf{J} </math> and <math> V, \rho </math> in [[electrostatics]]:<ref name="Semon1996">{{cite journal
 | author = Mark D. Semon and John R. Taylor
 | title = Thoughts on the magnetic vector potential
 | journal = American Journal of Physics
 | volume = 64
 | issue = 11
 | pages = 1361–1369
 | year = 1996
 | doi = 10.1119/1.18400
 | bibcode = 1996AmJPh..64.1361S
 | url = https://doi.org/10.1119/1.18400
| url-access = subscription
 }}</ref>
<math display="block"> \nabla^2 \mathbf{A} = -\mu_0 \mathbf{J} </math>
just like the electrostatic equation
<math display="block"> \nabla^2 V = -\frac{\rho}{\epsilon_0} </math> 

Likewise one can integrate to obtain the potentials:
<math display="block"> \mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4 \pi} \int_R  \frac{\mathbf{J}(\mathbf{r}')}{\left | \mathbf{r} - \mathbf{r}' \right |} \mathrm{d}^3 r'</math>
just like the equation for the [[electric potential]]:
<math display="block"> V(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \int_R \frac{\rho(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 r' </math>

== Interpretation as Potential Momentum ==
{{context|date=April 2025}}
By equating [[Newton%27s_laws_of_motion#Second_law|Newton's second law]] with the [[Lorentz force law]] we can obtain<ref name="Semon1996" />
<math display="block">
m\frac{\mathrm{d}v}{\mathrm{d}t}=q \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}   \right).</math>
Dotting this with the velocity  yields
<math display="block">
\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{1}{2}m v^2\right) = q\mathbf{v} \cdot \left(\mathbf{E} + \mathbf{v}\times\mathbf{B} \right).
</math>
With the [[dot product]] of the [[cross product]] being zero, substituting
<math display="block">\mathbf{E} =  - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t },</math>
and the [[ convective derivative]] of <math>\phi</math> in the above equation then gives
<math display="block"> \frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{1}{2} mv^2 + q \phi \right ) = \frac{\partial}{\partial t} q \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>
which tells us the time derivative of the "generalized energy" <math> \frac{1}{2} mv^2 + q \phi </math> in terms of a velocity dependent potential <math> q \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>, and
<math display="block"> \frac{\mathrm{d}}{\mathrm{d} t} \left ( mv + q \mathbf{A} \right ) = - \nabla q \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>
which gives the time derivative of the [[generalized momentum]] <math> m \mathbf{v} + q \mathbf{A} </math> in terms of the (minus) gradient of the same velocity dependent potential.

Thus, when the (partial) time derivative of the velocity dependent potential <math> q (\phi - \mathbf{v} \cdot \mathbf{A}) </math> is zero, the generalized energy is conserved, and likewise when the gradient is zero, the generalized momentum is conserved. As a special case, if the potentials are time or space symmetric, then the generalized energy or momentum respectively will be conserved. Likewise the fields contribute <math> q \mathbf{r} \times \mathbf{A} </math> to the generalized angular momentum, and rotational symmetries will provide conservation laws for the components. 

Relativistically, we have the single equation
<math display="block"> \frac{\mathrm{d}}{\mathrm{d} \tau} \left ( p^{\mu} + qA^{\mu} \right ) = \partial_{\nu} \left ( U^{\mu} \cdot A^{\mu} \right ) </math>
where
* <math> \tau </math> is the [[proper time]],
* <math> p^{\mu} </math> is the [[four momentum]] <math> (E/c, \gamma m \mathbf{v}) </math>
* <math> U^{\mu} </math> is the [[four velocity]] <math> \gamma (c, \mathbf{v}) </math>
* <math> A^{\mu} </math> is the [[four potential]] <math> (\phi/c, \mathbf{A}) </math>
* <math> \partial_{\nu} </math> is the [[four gradient]] <math> (\frac{\partial}{\partial \left ( ct \right )}, -\nabla) </math>

=== Analytical Mechanics of a Charged Particle ===
In a field with electric potential <math>\ \phi\ </math> and magnetic potential <math>\ \mathbf{A}</math>, the [[Lagrangian mechanics|Lagrangian]] (<math>\ \mathcal{L}\ </math>) and the [[Hamiltonian mechanics|Hamiltonian]] (<math>\ \mathcal{H}\ </math>) of a particle with mass <math>\ m\ </math> and charge <math>\ q\ </math> are<math display="block">\begin{aligned}
\mathcal{L} &= \frac{1}{2} m\ \mathbf v^2 + q\ \mathbf  v \cdot  \mathbf A - q\ \phi\ ,\\
\mathcal{H} &= \frac{1}{2m}\left( \mathbf{p} - q\mathbf A \right)^2 + q\ \phi ~.
\end{aligned}</math>

The generalized momentum <math> \mathbf{p} </math> is <math> \frac{\partial \mathcal{L}}{\partial v} = m \mathbf{v} + q \mathbf{A} </math>. The generalized force is <math> \nabla \mathcal{L} = -q \nabla  \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>. These are exactly the quantities from the previous section. It this framework, the conservation laws come from [[Noether's theorem]].

=== Example: Solenoid ===
Consider a charged particle of charge <math> q </math> located distance <math> r </math> outside a solenoid oriented on the <math> z </math> that is suddenly turned off. By [[Faraday's law of induction]], an electric field will be induced that will impart an impulse to the particle equal to <math> q \Phi_0/2 \pi r \hat{\phi} </math> where <math> \Phi_0 </math> is the  initial [[magnetic flux]] through a cross section of the solenoid. <ref> {{cite book | last1=Feynman | first1=Richard P. | author-link1=Richard Feynman | last2=Leighton | first2=Robert B. | author-link2=Robert B. Leighton | last3=Sands | first3=Matthew | author-link3=Matthew Sands | title=The Feynman Lectures on Physics | volume=2 | chapter=17 | chapter-url=https://www.feynmanlectures.caltech.edu/II_17.html | publisher=Addison-Wesley | year=1964 | isbn=978-0-201-02115-8 }} </ref>

We can analyze this problem from the perspective of generalized momentum conservation.<ref name=Semon1996 /> Using the analogy to Ampere's law, the magnetic vector potential is <math> \mathbf{A}(r) = \Phi_0/2 \pi r \hat{\phi} </math>. Since <math> \mathbf{p} + q\mathbf{A} </math> is conserved, after the solenoid is turned off the particle will have momentum equal to <math> q \mathbf{A} = q \Phi_0/2 \pi r \hat{\phi} </math>

Additionally, because of the symmetry, the <math> z </math> component of the generalized angular momentum is conserved. By looking at the [[Poynting vector]] of the configuration, one can deduce that the fields have nonzero total angular momentum pointing along the solenoid. This is the angular momentum transferred to the fields.

== Gauge choices ==
{{Main|Gauge fixing}}

The above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add [[Curl (mathematics)|curl]]-free components to the magnetic potential without changing the observed magnetic field. Thus, there is a [[degrees of freedom (physics and chemistry)|degree of freedom]] available when choosing <math>\mathbf{A}</math>. This condition is known as [[gauge invariance]].

Two common gauge choices are
* The [[Lorenz gauge]]: <math display="block">\ \nabla \cdot \mathbf{A} + \frac{1}{\ c^2} \frac{\partial \phi}{\partial t} = 0 </math>
* The [[Coulomb gauge]]: <math display="block">\ \nabla \cdot \mathbf{A} = 0 </math>

=== Lorenz gauge ===
{{see also|Mathematical descriptions of the electromagnetic field#Potential field approach|l1=Potential formulation of electromagnetic field}}
{{Main|Retarded potential}}

In other gauges, the formulas for <math>\mathbf{A}</math> and <math>\phi</math> are different; for example, see ''[[Coulomb gauge]]'' for another possibility.

==== Time domain ====
Using the above definition of the potentials and applying it to the other two Maxwell's equations (the ones that are not automatically satisfied) results in a complicated differential equation that can be simplified using the [[Lorenz gauge]] where <math>\mathbf{A} </math> is chosen to satisfy:<ref name=Feynman1515/>
<math display="block">\ \nabla \cdot \mathbf{A} + \frac{1}{\ c^2} \frac{\partial \phi}{\partial t} = 0 </math>

Using the Lorenz gauge, the [[electromagnetic wave equation]]s can be written compactly in terms of the potentials, <ref name=Feynman1515/>
* Wave equation of the scalar potential <math display="block">\begin{align}
              \nabla^2\phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\ \partial t^2} &= - \frac{\rho}{\epsilon_0} \\[2.734ex]
\end{align}</math>
* Wave equation of the vector potential <math display="block">\begin{align}
  \nabla^2\mathbf{A} - \frac{1}{\ c^2} \frac{\partial^2 \mathbf{A}}{\ \partial t^2} &= - \mu_0\ \mathbf{J}
\end{align}</math>

The solutions of Maxwell's equations in the Lorenz gauge (see Feynman<ref name=Feynman1515/> and Jackson<ref name=Jackson246>{{harvp|Jackson|1999|p=246}}</ref>) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the [[retarded potential]]s, which are the magnetic vector potential <math>\mathbf{A}(\mathbf{r}, t) </math> and the electric scalar potential <math>\phi(\mathbf{r}, t)</math> due to a current distribution of [[current density]] <math>\mathbf{J}(\mathbf{r}, t)</math>,  [[charge density]] <math>\rho(\mathbf{r}, t) </math>, and [[volume]] <math>\Omega </math>, within which <math>\rho</math> and <math>\mathbf{J}</math> are non-zero at least sometimes and some places):
* Solutions <math display="block">\begin{align}
  \mathbf{A}\!\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega \frac{ \mathbf{J}\left(\mathbf{r}' , t'\right) } R\ d^3\mathbf{r}' \\
        \phi\!\left(\mathbf{r}, t\right) &= \frac{1}{4\pi\epsilon_0} \int_\Omega \frac{ \rho \left(\mathbf{r}', t'\right) } R\ d^3\mathbf{r}'
\end{align}</math>
where the fields at [[position vector]] <math>\mathbf{r}</math> and time <math>t</math> are calculated from sources at distant position <math>\mathbf{r}' </math> at an earlier time <math>t' .</math> The location <math>\mathbf{r}'</math> is a source point in the charge or current distribution (also the integration variable, within volume <math>\Omega</math>). The earlier time <math>t'</math> is called the ''[[retarded time]]'', and calculated as
<math display="block"> R = \bigl\|\mathbf{r} - \mathbf{r}' \bigr\| ~.</math>
<math display="block"> t' = t - \frac{\ R \ }{c} ~.</math>

With these equations:
* The [[Lorenz gauge condition]] is satisfied:
: <math display="block">\ \nabla \cdot \mathbf{A} + \frac{1}{\ c^2}\frac{\partial\phi}{\partial t} = 0 ~.</math>
* The position of <math>\mathbf{r}</math>, the point at which values for <math>\phi</math> and <math>\mathbf{A}</math> are found, only enters the equation as part of the scalar distance from <math>\mathbf{r}'</math> to <math>\mathbf{r} .</math> The direction from <math>\mathbf{r}'</math> to <math>\mathbf{r} </math> does not enter into the equation. The only thing that matters about a source point is how far away it is.
* The integrand uses ''[[retarded time]]'', <math>t' .</math> This reflects the fact that changes in the sources propagate at the speed of light. Hence the charge and current densities affecting the electric and magnetic potential at <math>\mathbf{r}</math> and <math>t</math>, from remote location <math> \mathbf{r}' </math> must also be at some prior time <math>t'.</math>
* The equation for <math>\mathbf{A}</math> is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:<ref name=KrausE>{{harvp|Kraus|1984|p=189}}</ref> <math display="block">\begin{align}
  A_x\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega\frac{J_x\left(\mathbf{r}', t'\right)}R\ d^3\mathbf{r}'\ , \qquad
  A_y\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega\frac{J_y\left(\mathbf{r}', t'\right)}R\ d^3\mathbf{r}'\ ,\qquad
  A_z\left(\mathbf{r}, t\right) &= \frac{\mu_0}{\ 4\pi\ } \int_\Omega\frac{J_z\left(\mathbf{r}', t'\right)}R\ d^3\mathbf{r}' ~.
\end{align}</math> In this form it is apparent that the component of <math>\mathbf{A}</math> in a given direction depends only on the components of <math>\mathbf{J} </math> that are in the same direction. If the current is carried in a straight wire, <math>\mathbf{A}</math> points in the same direction as the wire.

==== Frequency domain ====

The preceding time domain equations can be expressed in the frequency domain.<ref name="Balanis_Ant">{{Citation |last=Balanis |first= Constantine A. |year= 2005 |title= Antenna Theory |edition=third |publisher= John Wiley |isbn= 047166782X |author-link=Constantine A. Balanis}}</ref>{{rp|139}}
* Lorenz gauge <math>\nabla \cdot \mathbf{A} + \frac{j \omega}{c^2} \phi= 0 \qquad </math> or <math>\qquad  \phi = \frac {j \omega}{k^2} \nabla \cdot \mathbf{A} </math>
* Solutions <math> \mathbf{A}\!\left(\mathbf{r}, \omega \right) = \frac{\mu_0}{\ 4\pi\ } \int_\Omega \frac{ \mathbf{J}\left(\mathbf{r}' , \omega \right) }R\  e^{-jkR} d^3 \mathbf{r}' \qquad
\phi\!\left(\mathbf{r}, \omega \right) = \frac{1}{4\pi\epsilon_0} \int_\Omega \frac{ \rho \left(\mathbf{r}', \omega \right) } R \ e^{-jkR} d^3 \mathbf{r}'</math>
* Wave equations <math> \nabla^2 \phi + k^2 \phi = - \frac{\rho}{\epsilon_0}  \qquad  \nabla^2 \mathbf{A} + k^2 \mathbf{A} = - \mu_0\ \mathbf{J} . </math>
* Electromagnetic field equations <math>\mathbf{B} = \nabla \times \mathbf{A}\ \qquad \mathbf{E} = -\nabla \phi - j \omega \mathbf{A}  = - j \omega \mathbf{A} -j \frac {\omega}{k^2} \nabla ( \nabla \cdot \mathbf{A} )</math>
where
: <math> \phi  </math> and <math> \rho </math> are scalar [[phasors]].
: <math> \mathbf{A},  \mathbf{B},  \mathbf{E}, </math> and <math> \mathbf{J} </math> are vector [[phasors]].
: <math> k = \frac \omega c </math>

There are a few notable things about <math>\mathbf{A}</math> and <math>\phi</math> calculated in this way:
* The [[Lorenz gauge condition]] is satisfied: <math>\textstyle \phi = -\frac {c^2}{j \omega}\nabla \cdot \mathbf{A}. </math>  This implies that the frequency domain electric potential, <math>\phi</math>, can be computed entirely from the current density distribution, <math>\mathbf{J}</math>.
* The position of <math>\mathbf{r},</math> the point at which values for <math>\phi</math> and <math>\mathbf{A}</math> are found, only enters the equation as part of the scalar distance from <math>\mathbf{r}'</math> to <math>\ \mathbf{r}.</math> The direction from <math>\mathbf{r}'</math> to <math>\mathbf{r}</math> does not enter into the equation. The only thing that matters about a source point is how far away it is.
* The integrand uses the [[phase shift]] term <math> e^{-jkR} </math> which plays a role equivalent to ''[[retarded time]]''. This reflects the fact that changes in the sources propagate at the speed of light; propagation delay in the time domain is equivalent to a phase shift in the frequency domain.  
* The equation for <math>\mathbf{A}</math> is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:<ref name=KrausE>{{harvp|Kraus|1984|p=189}}</ref> <math display="block">\begin{align}
        \mathbf{A}_x\!\left(\mathbf{r}, \omega \right) &= \frac{\mu_0}{4\pi} \int_\Omega \frac{ \mathbf{J}_x \left(\mathbf{r}' , \omega \right) }R\  e^{-jkR} \ d^3\mathbf{r}', 
\qquad  \mathbf{A}_y\!\left(\mathbf{r}, \omega \right) &= \frac{\mu_0}{4\pi} \int_\Omega \frac{ \mathbf{J}_y \left(\mathbf{r}' , \omega \right) }R\  e^{-jkR} \ d^3\mathbf{r}',   
\qquad  \mathbf{A}_z\!\left(\mathbf{r}, \omega \right) &= \frac{\mu_0}{4\pi} \int_\Omega \frac{ \mathbf{J}_z \left(\mathbf{r}' , \omega \right) }R\  e^{-jkR} \ d^3\mathbf{r}' 
\end{align}</math> In this form it is apparent that the component of <math>\mathbf{A}</math> in a given direction depends only on the components of <math>\ \mathbf{J}\ </math> that are in the same direction. If the current is carried in a straight wire, <math>\mathbf{A}</math> points in the same direction as the wire.

== Depiction of the A-field ==
[[File:Magnetic Vector Potential Circular Toroid.svg|right|450px|thumb|Representing the [[Coulomb gauge]] magnetic vector potential <math>\mathbf{A}</math>, magnetic flux density <math>\mathbf{B}</math> and current density <math>\mathbf{J}</math> fields around a [[toroidal inductor]] of circular [[Cross section (geometry)|cross section]]. Thicker lines ,indicate field lines of higher average intensity. Circles in the cross section of the core represent the <math>\ \mathbf{B}</math> field coming out of the picture, plus signs represent <math>\mathbf{B}</math> field going into the picture. <math>\nabla\cdot\mathbf{A} = 0</math> has been assumed.]]

See Feynman<ref name=Feynman1511>{{harvp|Feynman|1964|loc=[https://feynmanlectures.caltech.edu/II_15.html cpt 15]|p=11}}</ref> for the depiction of the <math>\mathbf{A}</math> field around a long thin [[solenoid]].

Since 
<math display="block">\nabla \times \mathbf{B} = \mu_0\ \mathbf{J}</math>
assuming quasi-static conditions, i.e. 
: <math>\frac{\ \partial\mathbf{E}\ }{\partial t} \to 0\ </math> and <math>\ \nabla \times \mathbf{A} = \mathbf{B}</math>,
the lines and contours of <math>\ \mathbf{A}\ </math> relate to <math>\ \mathbf{B}\ </math> like the lines and contours of <math>\mathbf{B}</math> relate to <math>\ \mathbf{J} .</math> Thus, a depiction of the <math>\mathbf{A}</math> field around a loop of <math>\mathbf{B}</math> flux (as would be produced in a [[toroidal inductor]]) is qualitatively the same as the <math>\mathbf{B}</math> field around a loop of current.

The figure to the right is an artist's depiction of the <math> \mathbf{A} </math> field. The thicker lines indicate paths of higher average intensity (shorter paths have higher intensity so that the path integral is the same). The lines are drawn to (aesthetically) impart the general look of the {{nowrap|<math> \mathbf{A}</math> field.}}

The drawing tacitly assumes <math> \nabla \cdot \mathbf{A} = 0</math>, true under any one of the following assumptions:
* the [[Coulomb gauge]] is assumed
* the [[Lorenz gauge]] is assumed and there is no distribution of charge, <math>\rho = 0</math> 
* the [[Lorenz gauge]] is assumed and zero frequency is assumed
* the [[Lorenz gauge]] is assumed and a non-zero frequency, but still assumed sufficiently low to neglect the term <math>\textstyle \frac{1}{c} \frac{\partial\phi}{\partial t} </math>

{{clear}}
== Electromagnetic four-potential ==
{{Main|Electromagnetic four-potential}}

In the context of [[special relativity]], it is natural to join the magnetic vector potential together with the (scalar) [[electric potential]] into the [[electromagnetic potential]], also called ''four-potential''.

One motivation for doing so is that the four-potential is a mathematical [[four-vector]]. Thus, using standard four-vector transformation rules, if the electric and magnetic potentials are known in one inertial reference frame, they can be simply calculated in any other inertial reference frame.

Another, related motivation is that the content of classical electromagnetism can be written in a concise and convenient form using the electromagnetic four potential, especially when the [[Lorenz gauge]] is used. In particular, in [[abstract index notation]], the set of [[Maxwell's equations]] (in the Lorenz gauge) may be written (in [[Gaussian units]]) as follows:
<math display="block">\begin{align}
  \partial^\nu A_\nu &= 0 \\
          \Box^2 A_\nu &= \frac{ 4\pi }{\ c\ }\ J_\nu
\end{align}</math>
where <math>\ \Box^2\ </math> is the [[d'Alembertian]] and <math>\ J\ </math> is the [[four-current]]. The first equation is the [[Lorenz gauge condition]] while the second contains Maxwell's equations.  The four-potential also plays a very important role in [[quantum electrodynamics]].

== See also ==
* [[Magnetic scalar potential]]
* [[Aharonov–Bohm effect]]
* [[Gluon field]]

== Notes ==
{{reflist|2}}

== References ==
* {{cite book
 | last = Duffin
 | first = W.J.
 | title = Electricity and Magnetism, Fourth Edition
 | publisher = McGraw-Hill
 | year = 1990
}}
* {{cite book
 |last1= Feynman
 |first1= Richard P
 |last2= Leighton
 |first2= Robert B
 |last3= Sands
 |first3= Matthew
 |year= 1964
 |title= The Feynman Lectures on Physics Volume 2
 |publisher= Addison-Wesley
 |isbn= 0-201-02117-X
 |ref = {{harvid|Feynman|1964}}
 |url = https://feynmanlectures.caltech.edu/II_toc.html
}}
* {{cite book
 |last=Jackson
 |first= John David
 |year= 1999
 |title= Classical Electrodynamics
 |edition= 3rd
 |publisher= [[John Wiley & Sons]]
 |isbn= 0-471-30932-X
}}
* {{cite book
 |last= Kraus
 |first= John D.
 |year= 1984
 |title= Electromagnetics
 |edition= 3rd
 |publisher= McGraw-Hill
 |isbn= 0-07-035423-5
 |url-access= registration
 |url= https://archive.org/details/electromagnetics00krau
}}

== External links ==
* {{Commons category inline}}

[[Category:Potentials]]
[[Category:Magnetism]]
[[Category:Vector physical quantities]]