Who discovered the magnetic vector potential, $\vec{A}$?

Question

Neither Maxwell’s fundamental differential equations on electromagnetism nor Einstein’s first papers considered the magnetic vector potential A. So who discovered, formulated or used A for the first time?

Answer 1

Franz Ernst Neumann was the first¹ to write down the magnetic vector potential in his 1845 paper "General laws of induced electrical currents." He used it to write the equation summarizing Faraday's induction experiment (Faraday's law).

The original paper:

1. F. E. Neumann, “Allgemeine Gesetze Der Inducirten Elektrischen Ströme,” Annalen Der Physik 143, no. 1 (January 1, 1846): 31–44, doi:10.1002/andp.18461430103.

Neumann's Dictionary of Scientific Biography entry says this:

Neumann and his contemporary Wilhelm Weber were the founders of the electrodynamic school in Germany, which later included, among others, Riemann, Betti, Carl Neumann, and Lorenz. The investigations and analyses of this group were guided by the assumption, held originally by Ampère, that electromagnetic phenomena resulted from direct action at a distance rather than through the mediation of a field. Neumann’s major contributions were contained in two papers published in 1845 and 1848, in which he established mathematically the laws of induction of electric currents. The papers, transmitted to the Berlin Academy, were entitled “Allgemeine Gesetze der inducirten elektrischen Ströme” and “Ober ein allgemeines Princip der mathematischen Theorie inducirter elektrischer Ströme.”

As a starting point Neumann took the proposition, formulated in 1834 by F. E. Lenz after Faraday’s discovery of induction, that the current induced in a conductor moving in the vicinity of a galvanic current or a magnet will flow in the direction that tends to oppose the motion. In his mathematical analysis Neumann arrived at the formula $E.Ds =—ν C.Ds$, where $Ds$ is an element of the moving conductor, $E.Ds$ is the elementary induced electromotive force, $v$ is the velocity of the motion, $C.Ds$ is the component of the inducing current, and $∊$ is a constant coefficient. With this formula Neumann was able to calculate the induced current in numerous particular instances. At present a common formulation is $E = — dN/dt$, where $E$ is the electromotive force generated in the circuit through which the number of magnetic lines of force is changing at the rate of $dN/dt$.

Continuing his analysis Neumann noticed a way in which the treatment of currents induced in closed circuits moving in what is now termed a magnetic field might be generalized. He saw that the induced current depends only on the alteration, caused by the motion, in the value of a particular function. Considering Ampère’s [force] equations for a closed circuit, Neumann arrived at what is known as the mutual potential of two circuits, that is, the amount of mechanical work that must be performed against the electromagnetic forces in order to separate the two circuits to an infinite distance apart, when the current strengths are maintained unchanged. In modern notation the potential function, $Vii'$, is written:$$Vii'=-ii'\iint\frac{\mathbf{ds}\cdot\mathbf{ds'}}{r}.$$

Continuing his analysis Neumann noticed a way in which the treatment $\mathbf{ds}.\mathbf{ds'}$ is the scalar product of the two vectors $\mathbf{ds}$ and $\mathbf{ds'}$, and $r$ their distance apart. If a fixed element $\mathbf{ds'}$ is taken and integrated with respect to $\mathbf{ds}$, the vector potential of the first circuit at the point occupied by $\mathbf{ds}$ is obtained. Maxwell arrived at the concept of vector potentials by another method and interpreted them as analytical measures of Faraday’s electrotonic state.

---

References

1. André Koch Torres Assis, Weber’s Electrodynamics (Dordrecht; Boston: Kluwer Academic, 1994), §5.2 ("Franz Neumann"), pp. 122-5.

2. Maxwell used the "magnetic vector potential" (what he called the "electromagnetic momentum") in his A Dynamical Theory of the Electromagnetic Field (Scientific Works vol. 1 p. 555-556). $\mathbf{E}$ is $(P,Q,R)$ and $\mathbf{A}$ is $(F,G,H)$ in Maxwell's notation.

Answer 2

About vector potential there are different situations,

(1) Neumann vector potential $\boldsymbol{A}_{N}$(1845):

$ \boldsymbol{A}_{N}=\frac{\mu_{0}}{4\pi}\iiint_{V}\frac{\boldsymbol{J}(x',t)}{r}dV $

$ r=|\boldsymbol{x}-\boldsymbol{x}'| $

This formula should be credit to Neumann. Perhaps Neumann did not written the above formula, but Neumann has the formula,

$ U=II'\oint\oint\frac{d\boldsymbol{l}\cdot d\boldsymbol{l}'}{r} $

$U$ is the energy move the two coils from distance $r$ to infinity. From the above it is easy to write,

$ \boldsymbol{A}=\oint\frac{I'}{r}d\boldsymbol{l}' $

Hence

$ U=\oint\boldsymbol{A}\cdot d\boldsymbol{l} $

(2)Weber vector potential $\boldsymbol{A}_{W}$(1846)

$ \boldsymbol{A}_{W}=\frac{\mu_{0}}{4\pi}\iiint_{V}\frac{(\boldsymbol{J}(x',t)\cdot\boldsymbol{r})\boldsymbol{r}}{r}dV $

$ \boldsymbol{r}=\boldsymbol{x}-\boldsymbol{x}' $

(3) William Thomson (Lord Kelvin) 1847

$ \boldsymbol{B}=\nabla\times\boldsymbol{A} $

Where $\boldsymbol{A}$ is Neumann or Weber vector potential.

$ \boldsymbol{B}=\frac{\mu_{0}}{4\pi}\iiint_{V}\boldsymbol{J}(x',t)\times\frac{\boldsymbol{r}}{r^{3}}dV $

$ =\frac{\mu_{0}}{4\pi}\iiint_{V}\nabla(\frac{1}{r})\times\boldsymbol{J}(x',t)dV $

$ =\frac{\mu_{0}}{4\pi}\iiint_{V}\nabla\times\frac{\boldsymbol{J}(x',t)}{r}dV $

$ =\nabla\times\boldsymbol{A} $

(4) Maxwell (1856)

$ \boldsymbol{E}=-\frac{\partial\boldsymbol{A}_{N}}{\partial t} $

(5) Gustav Robert Kirchhoff (1857)

$ \boldsymbol{J}=-\sigma(\nabla\phi+\frac{\partial\boldsymbol{A}_{W}}{\partial t}) $

(6) Gustav Robert Kirchhoff gauge condition 1857,

$ \nabla\cdot\boldsymbol{A}_{W}=\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\phi $

$ \phi(\boldsymbol{x},t)=\frac{1}{4\pi\epsilon_{0}}\iiint_{V}\frac{\rho(\boldsymbol{x}',t)}{r}dV $

Kirchhoff got his result by using the current continuity equation,

$ \nabla\cdot\boldsymbol{J}=-\frac{\partial}{\partial t}\rho $

Continuity equation is very important, it was the reason Maxwell introduced the displacement current

$ \frac{\partial\boldsymbol{D}}{\partial t} $

(7) Retarded potential Lorenz (1867)

$ \boldsymbol{A}=\frac{\mu_{0}}{4\pi}\iiint_{V}\frac{\boldsymbol{J}(x',t-r/c)}{r}dV $

$ \phi(\boldsymbol{x},t)=\frac{1}{4\pi\epsilon_{0}}\iiint_{V}\frac{\rho(x',t-r/c)}{r}dV $

$c$ is the speed of light. In the same time Lorenz offers the Lorenz gauge condition.

(8) Lorenz gauge condition. (1867)

$ \nabla\cdot\boldsymbol{A}_{N}=-\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\phi $

Kirchhoff gauge condition (1857) derivation has shown that

$ \nabla\cdot\boldsymbol{A}_{N}-\nabla\cdot\boldsymbol{A}_{W}=-2\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\phi $

Lorenz get the Lorenz gauge condition is too easy. I thought Lorenz gauge perhaps should be referred as Kirchhoff-Lorenz gauge. Any way Lorenz gauge is very important. It is a is only bridge to get correct retarded potential and advanced potential (for both vector and scale potential).

Summary:

1. Neumann and Weber vector potential are the most important work to contributed for Maxwell equations, but even in the Maxwell equations this work has been credited to as Faraday induction law.
2. Lorenz gauge should be referred as Kirchhoff-Lorenz gauge. Kirchhoff did most work for Lorenz gauge. Without Lorenz gauge it is not possible to obtained correct solution of Maxwell's equations (retarded potential and advanced potential)
3. Lorenz retarded potential is equivalent to the theorem of Maxwell (retarded potential = displacement current). He follows the Kirchhoff's thought, however he has replaced the Weber vector potential as Neumann vector potential and obtained the important results of the retarded potential, which is the solution of Maxwell's equations.
4. If Kirchhoff had applied Neumann vector potential in 1857 instead of Weber vector potential, He would replace Maxwell as the greatest scientist in electromagnetic field theory. Kirchhoff first utilized the current continuity equation to obtained his gauge condition. Lorenz following Kirchhoff and got the correct results, but a few years later than Maxwell. Any way, Kirchhoff together with Lorenz, should get the same respect as Maxwell.