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I know that the electric field (a vector field) is the result of the gradient of the electric potential,which is a scalar field of the type: $\Phi$ : $\mathbb{R}^3 \rightarrow \mathbb{R}$. So the gradient will point towards the direction of the largest change in value.

But what about the magnetic vector potential, which is a vector field, whose curl gives us a magnetic vector field? How can I geometrically understand what the curl of the magnetic vector potential does, so the result is the magnetic field?

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    $\begingroup$ If you imagine the vector field as representing fluid velocity, the magnitude of its curl at any point is the rate that a small piece of fluid at that point would be rotating. The direction of the curl is along the axis of that rotation. $\endgroup$
    – Ben51
    Feb 8, 2021 at 0:13
  • $\begingroup$ the magnitude is the rate of rotation, what does that mean? rate of rotation. Do you have like a link for a visual representation. And how is the electric vector potential related to the magnetic potential, explained or represented visually, that way i can understand the geometry behind what we are doing, and not just math $\endgroup$
    – imbAF
    Feb 8, 2021 at 0:17
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    $\begingroup$ rate of rotation: radians per second or whatever. how fast the orientation is changing. $\endgroup$
    – Ben51
    Feb 8, 2021 at 0:21
  • $\begingroup$ Due diligence. $\endgroup$ Feb 8, 2021 at 1:04
  • $\begingroup$ The vector potential is roughly proportional to the current density, especially if you have a line current. The magnetic field is a circle centered on that line of current with it's axis parallel tot he current, i.e. $\vec{B} = \frac{\mu_0I}{2\pi r}\hat{\theta}$So the magnetic field literally curls around the vector potential (in $\hat{k}$ direction) in that case. For a more complicated current distribution, the effect is a combination of curls around microscopic lien charges. $\endgroup$
    – R. Romero
    Jan 25 at 23:12

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Warning: Seems like I got the action of the Lorentz force backward. I'll edit this post when I have the time.

To understand why we use the curl, it is best to review what the magnetic field is actually doing.

Principle behind magnetic field (and thus vector potential)

In the magnetostatic case, the vector potential represents the principle that moving charges tend to curve the trajectories of other moving charges, in a manner similar to a ball swirling in a stream. Since the magnetic field actually does not impart kinetic energy onto the charges it accelerates, I reiterate that the charges only curve their trajectories in a magnetic field.

Note that when I use the terms "drag" and "swirl", realize I am only applying these terms loosely. (Also, thanks to @Ben51 for reminding me of the stream analogy in a comment to OP's question.)

The magnetic field points in the direction about which that the charges would swirl (using right hand rule for positive charge, and left hand rule for negative charge), and its magnitude represents the strength of the swirling effect. This is why the curl appears in the magnetic contribution to the Lorentz force.

In certain gauges, the magnetic vector potential will point in the direction of the original current that sources it, but decay inversely with distance. (This intuitively shows that when finding the magnetic field, we care about moving charges, and that such an effect decays with distance.)

Because the magnetic field is really what we care about, we have gauge freedom in the vector potential. This gauge freedom represents that we only care about the part of the vector potential that impacts the swirling principle; we ignore the rest. (Saying $\mathbf{B}=\nabla\times\mathbf{A}$ is equivalent is to saying that we don't believe in the existence of magnetic monopoles.)

We're going to examine an explanatory situation: the magnetic field around an infinite wire. Besides the definition of magnetic vector potential $\nabla\times\mathbf{A}=\mathbf{B}$, let's keep in mind two other equations as we approach this situation.

The Lorentz Force

$\mathbf{F} = q\mathbf{E} + q\mathbf{v}\times \mathbf{B}$

The Lorentz force, combined with Maxwell's equations (and Newton's second law $\mathbf{F}=m\mathbf{a})$, explains how charges affect the motion of other charges, through the use of the concepts of the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$.

The important thing to note is that magnetic fields don't push or pull on a charge, but only deflect the motion of a charge in a way that causes it to curve in space. In a uniform magnetic field, the curved trajectories become circular orbits.

Retarded vector potential from a current density

This equation, which comes from the solutions to Maxwell's equations in the Lorenz gauge, takes the form $$ \begin{aligned}\mathbf {A} \left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} \left(\mathbf {r} ',t'\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\,\mathrm {d} ^{3}\mathbf {r}\end{aligned}.$$

What is important to note is that it is in the same direction as the current flow, takes time to emanate from it (this is what the retarded time $t'$ represents), and decays with distance. So it is very intuitive to interpret this equation.

Explanatory Example

Imagine an infinite wire with a current $I$.

Diagram of vector potential about a wire.

Note the vector potential takes the same direction as the current. A positive charge moving near to this wire with an initial velocity $\vec{v}_0$ curves its trajectory in the produced field. The decay of the vector potential with distance represents a "dragging effect" that decays with distance; a moving charges only feels the differential of this effect, and curves its trajectory. I find this similar to explanations of refraction: light might curve when the index of refraction varies in space. But in the case of magnetic field, the amount that the "left side" of a charge is sped up and the "right side" of charge is slowed down exactly balances to curve the trajectory, never to speed up or slow down the charge.

I hope you gain some intuition as to why the vector potential is defined the way it is, what it is really doing. Crucially, in the Lorenz gauge, note currents only source vector potentials aligned with the sourcing current; of course, you can always use another gauge if you find that more intuitive.

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