On Physical Lines of Force/Modern derivations/The vector potential

On Physical Lines of Force → Modern derivations → The vector potential

What Maxwell had

In Prop. VIII of Part II, Maxwell asks what quantity, differentiated, yields the magnetic force. He introduces three functions $F,G,H$ and writes equation (55):

$$\mu \alpha =\frac{dH}{dy}-\frac{dG}{dz},\mu \beta =\frac{dF}{dz}-\frac{dH}{dx},\mu \gamma =\frac{dG}{dx}-\frac{dF}{dy}\mathrm{(}\mathrm{55}\mathrm{)}$$

Provenance: the equation number (55) is attested; the body is a reconstruction in Maxwell's notation. See Sources.

In modern notation those three lines are one:

$$𝐁=\mathrm{\nabla }\times 𝐀$$

where $𝐀=(F,G,H)$ is the vector potential. Take care with the letters: Maxwell's $H$ here is the third component of $𝐀$, not the magnetic field. See Notation.

The operation on the right is the curl. Where the divergence measures how much a field spreads out of a point, the curl measures how much it circulates around one: $\mathrm{\nabla }\times 𝐀$ is the vector whose component along any direction gives the circulation of $𝐀$ round a small loop facing that way, per unit area enclosed. Maxwell's three lines are its three components.

Why this is always possible

That $𝐁$ can be written this way is not an accident of the vortex model. It is equivalent to the statement that magnetic lines of force have no ends.

One direction is immediate. The divergence of a curl vanishes identically — write it out and the six second derivatives cancel in pairs. So any $𝐁$ of the form $\mathrm{\nabla }\times 𝐀$ automatically satisfies $\mathrm{\nabla }\cdot 𝐁=0$. This is exactly how Maxwell obtains equation (56), his statement that there are no magnetic sources: it costs him nothing, because it was already true of (55).

The converse also holds, in any region without holes in it — all of space, for instance. Given a field with $\mathrm{\nabla }\cdot 𝐁=0$, here is a vector potential for it, built by integration:

$$A_x={\int }_0^z{B}_y(x,y,z^{\prime })d{z}^{\prime },A_y=-{\int }_0^z{B}_x(x,y,z^{\prime })d{z}^{\prime }+{\int }_0^x{B}_z(x^{\prime },y,0)d{x}^{\prime },A_z=0$$

Differentiating gives $B_x$ and $B_y$ back at once; the third component works only after $\partial B_x/\partial x+\partial B_y/\partial y$ is replaced by $-\partial B_z/\partial z$, which is where the assumption is spent. So $𝐁=\mathrm{\nabla }\times 𝐀$ and $\mathrm{\nabla }\cdot 𝐁=0$ say the same thing in two ways, and choosing between them is a matter of what you want to do next.

Gauge freedom

The construction above set $A_z=0$ for convenience, which should make you suspect that $𝐀$ is not unique. It is not. Let $\chi$ be any scalar function of position and time and replace

$$𝐀⟶𝐀+\mathrm{\nabla }\chi$$

Then $𝐁$ does not change, because the curl of a gradient is identically zero, just as the divergence of a curl is. Nothing measurable has altered, and yet $𝐀$ is a different function. This is gauge freedom.

There is a catch, and it is worth spelling out because it is where the two potentials are locked together. The electric field is written in terms of both, in Maxwell's equation (58):

$$𝐄=-\frac{\partial 𝐀}{\partial t}-\mathrm{\nabla }\phi$$

Change $𝐀$ by $\mathrm{\nabla }\chi$ and $𝐄$ acquires an unwanted $-\mathrm{\nabla }(\partial \chi /\partial t)$. It is cancelled exactly if the scalar potential is changed at the same time, $\phi \to \phi -\partial \chi /\partial t$. Make both changes together and every field is untouched.

Since the choice is free, one makes it to suit the problem. Two are standard:

• Coulomb gauge, $\mathrm{\nabla }\cdot 𝐀=0$. Then $\phi$ obeys Poisson's equation and is the ordinary electrostatic potential of the instantaneous charge distribution. Convenient in statics and in bound-state atomic physics.
• Lorenz gauge, $\mathrm{\nabla }\cdot 𝐀+\frac{1}{c}\partial \phi /\partial t=0$. Then $𝐀$ and $\phi$ each satisfy the same wave equation, with sources. Convenient for radiation, and manifestly compatible with relativity. It is named for Ludvig Lorenz, a different physicist from the H. A. Lorentz of the force law; the near-collision of names is one of the standing traps of the subject.

Maxwell does not discuss any of this. His $F,G,H$ are fixed by the mechanics of the medium, and the question of what else they might have been does not arise. Gauge freedom is a fact about the equations, not about the world, and it took the equations being separated from the model before anyone needed to say so.

The electrotonic state, and why "momentum"

Maxwell identifies $F,G,H$ with Faraday's electrotonic state — the name Faraday gave to a condition of a body under induction that he was sure existed and could not characterize. Maxwell also calls the same triple the electromagnetic momentum of the medium, and that second name is the one to think with.

Compare two statements. In ordinary mechanics, force is the rate of change of momentum, $𝐅=d𝐩/dt$. In equation (58), electromotive force is the rate of change of $𝐀$, give or take the gradient term that belongs to electrostatics. Whatever $𝐀$ is, it stands to electromotive force as momentum stands to force. On the vortex model that is not an analogy but a mechanical identity: the vortices carry angular momentum, and $𝐀$ is the bookkeeping for it.

The name has aged well. In the Lagrangian mechanics of a charge $q$ in a field, the momentum conjugate to position is not $m𝐯$ but

$${𝐩}_{\text{can}}=m𝐯+\frac{q}{c}𝐀$$

and it is this quantity, not $m𝐯$, that is conserved when the system has a symmetry. Maxwell's word was chosen for a picture that has since been discarded, and it remains the right word.

Is A physical?

For a long time the standard answer was no: $𝐀$ is a convenience, a function you differentiate to get the real thing. Gauge freedom seems to settle it — a quantity you can change at will without changing any measurement is hard to call real.

Maxwell's medium says otherwise, and the anchor ring of Part II is where it says it. Wind a toroid evenly with wire. The magnetic lines close entirely inside the ring; outside, no magnet can detect anything. Yet a wire encircling the ring feels an electromotive force whenever the current in the coil varies.

The anchor ring: induction without any external field

This experiment shows that, in order to produce the electromotive force, it is not necessary that the conducting wire should be placed in a field of magnetic force, or that lines of magnetic force should pass through the substance of the wire or near it. All that is required is that lines of force should pass through the circuit of the conductor, and that these lines of force should vary in quantity during the experiment.

$𝐁$ is genuinely zero at the wire. $𝐀$ is not.

What it says today

The modern successor is the Aharonov–Bohm effect, proposed in 1959 and since confirmed. Send an electron beam past a long, perfectly shielded solenoid so that part of the beam goes one side and part the other, and let them interfere. Everywhere the electrons travel, $𝐁=0$. Nevertheless the interference pattern shifts, by a phase $\Delta \theta$ — not to be confused with the potential $\phi$ — of

$$\Delta \theta =\frac{q}{\hslash c}\oint 𝐀\cdot d\ell =\frac{q\Phi }{\hslash c}$$

the second equality being Stokes' theorem, which converts a circulation round a loop into the flux of the curl through any surface spanning it — here, the magnetic flux $\Phi$ threading the solenoid.

Now be careful about what this shows. It shows that the enclosed flux is physically detectable from outside the region where the field is, by a measurement made entirely in field-free space. It is often described as showing that the vector potential is physical, and there is something to that; but the quantity in the formula is gauge-invariant. Change $𝐀$ by $\mathrm{\nabla }\chi$ and the loop integral is unchanged, because the circulation of a gradient round a closed path is zero. What is measurable is the line integral of $𝐀$ round the circuit — equivalently the flux — and not $𝐀$ at any point.

Maxwell had no quantum mechanics, no electron, and no notion of a probability amplitude carrying a phase. The resemblance between his ring and the 1959 experiment is one of shape, not of content: two arrangements in which the action is delivered to a place the field never reaches, offered for the same reason — to show that the action is in the space, not at a distance.

Where the conventions differ

	Maxwell 1861	modern
the symbol	$F,G,H$, three components	$𝐀$, one vector
the name	electrotonic state; electromagnetic momentum	vector potential
the letter H	third component of $𝐀$	the magnetic field $𝐇$
the relation	$\mu \alpha =dH/dy-dG/dz$ and two more	$𝐁=\mathrm{\nabla }\times 𝐀$
derivatives	total, $d/dx$	partial, $\partial /\partial x$
gauge freedom	not raised; $F,G,H$ fixed by the mechanics	explicit, and a choice rather than a fact
the scalar	$\psi$, "electric tension"	$\phi$, the scalar potential
status of A	a real state of a real medium	gauge-dependent; its loop integral is not

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In the paper: Prop. VIII · Induced currents and the anchor ring