Editing Magnetic vector potential (section)

==== 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.