Editing Magnetic vector potential (section)

== 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}}