Magnetic Vector Potential

Two articles ago, $\nabla\cdot\vec B=0$ came with a structural promise attached: a field with no divergence is always somebody's curl. This article collects. Define $\vec A$ by $\vec B=\nabla\times\vec A$ and one of magnetostatics' two field equations is solved identically — not for some current distribution, for all of them — because the divergence of a curl vanishes as a matter of algebra. That is the same trick electrostatics played when $\nabla\times\vec E=0$ let us write $\vec E=-\nabla V$ and never think about the curl equation again. Magnetism's conserved statement is a divergence, so magnetism's potential is a vector: three components, curl instead of gradient, and a new kind of freedom in choosing it. Why bother? Three reasons, in ascending order of depth. Practically, $\vec A$ points along the current that makes it — it inherits the source's geometry the way $V$ piles up near charge — so it is often easier to compute than the wrap-around $\vec B$ , and one curl at the end buys the field. Structurally, $\vec A$ has a direct physical reading: its line integral around any loop is the magnetic flux through it, $\oint\vec A\cdot\mathrm{d}\vec\ell=\Phi_B$ , which makes $\vec A$ a kind of flux bookkeeper stationed along every curve in space. And fundamentally, the twentieth century ruled that $\vec A$ is not a scaffold: it enters the canonical momentum $\vec p=m\vec v+q\vec A$ , and a quantum charge circling a solenoid picks up a measurable phase from $\vec A$ in a region where $\vec B$ is exactly zero — the Aharonov–Bohm effect. One honest caveat runs through the article: $\vec A$ is not unique. Adding any gradient $\nabla\lambda$ changes nothing physical. Here we exploit only the simplest consequence; the full story of that freedom — what it means, how to spend it, and how it turns Ampère's law into a Poisson equation — is the next article's entire business.

Key takeaways

$\vec B=\nabla\times\vec A$ solves $\nabla\cdot\vec B=0$ identically (divergence of curl $\equiv0$ ); conversely, the Helmholtz machinery guarantees every divergence-free field has such an $\vec A$ .
$\vec A$ is determined only up to a gradient: $\vec A\to\vec A+\nabla\lambda$ leaves $\vec B$ untouched — gauge freedom, owned in full by the next article.
Flux meaning: $\oint\vec A\cdot\mathrm{d}\vec\ell=\Phi_B$ through the loop, by Stokes — gauge-invariant, and nonzero even where $\vec B=0$ (outside a solenoid $A=\Phi_B/2\pi s$ ).
For steady localized currents, $\vec A(\vec r)=\frac{\mu_0}{4\pi}\int\frac{\vec J(\vec r')}{\mathfrak{r}}\,\mathrm{d}\tau'$ — the exact transcription of the electrostatic potential integral, one for each Cartesian component; its curl reproduces Biot–Savart, its divergence vanishes for steady currents.
$\vec A$ points along its source current and is often the easier computation; infinite sources need the same reference-point care as the line charge's logarithmic $V$ .
Beyond statics, $\vec A$ is fundamental: canonical momentum $\vec p=m\vec v+q\vec A$ , the Aharonov–Bohm phase $q\Phi_B/\hbar$ , and the four-potential of relativity.
KEY IDEA: writing $\vec B=\nabla\times\vec A$ spends the no-monopole law once and for all, converts magnetostatics into an electrostatics-shaped problem, and costs only a gauge ambiguity that no measurable quantity ever feels.

The core insight

Solving $\nabla\cdot\vec B=0$ structurally by writing $\vec B=\nabla\times\vec A$ converts the coupled magnetostatic equations into a single, electrostatics-like problem for $\vec A$ , at the price of a gauge ambiguity that you fix by a convenient condition.

Where this lives today

The Aharonov–Bohm effect graduated from thought experiment to precision tool. Tonomura's 1986 electron-holography experiment — toroidal ferromagnets clad in superconducting niobium so that flux could neither leak nor be touched — showed the interference shift surviving with $\vec B$ rigorously excluded from the electron's path, settling a thirty-year argument: the potentials, not the local fields, couple to quantum phase. Today AB rings are standard instruments in mesoscopic physics — conductance through a nanoscale ring oscillates with period $h/e$ in the threaded flux, making $\oint\vec A\cdot\mathrm{d}\vec\ell$ a routinely dialed experimental knob — and flux quantization in superconducting loops ( $\Phi=nh/2e$ , the phase-coherence condition on the pair wavefunction) underpins SQUID magnetometers and every flux qubit. On the theory side, this article's modest bookkeeping device turned out to be the template for fundamental physics. The statement "physics depends on $\vec A$ only through gauge-invariant combinations" — promoted from observation to principle and generalized beyond electromagnetism — generates the Standard Model: demanding local phase freedom for matter fields forces gauge fields with exactly $\vec A$ 's transformation law into existence. The next article begins that story where it starts, with $\lambda$ .

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Edinburgh, Cambridge, and the Treatise years

Maxwell inherited $\vec A$ twice over — from Franz Neumann's induction theory (1845), where the object first appeared as a calculational potential, and from Faraday's "electrotonic state," a physical intuition without mathematics: Faraday was convinced that a wire near a magnet sat in a special condition, undetectable at rest, revealed as an induced current the instant anything changed. In "On Faraday's Lines of Force" (1856) Maxwell identified the two: his electrotonic intensity was a vector field whose curl gave the magnetic field and whose time derivative gave the induced electric force. By the Treatise (1873) he had renamed it the electromagnetic momentum — momentum per unit charge stored in the field configuration, cashed out as an impulse when currents switch — precisely the canonical-momentum reading modern mechanics assigns to $q\vec A$ . History then zigzagged: Heaviside and Hertz stripped the potentials from Maxwell's equations as "metaphysics," recasting everything in $\vec E$ and $\vec B$ — and quantum mechanics reinstated $\vec A$ at the foundation, exactly where Maxwell's instinct had put it.

History

1845

— Franz Neumann's induction theory introduces the vector potential as a mathematical device for computing induced EMFs.

1856

— Maxwell's "On Faraday's Lines of Force" gives Faraday's electrotonic state exact form: a field whose curl is B.

1873

— Maxwell's Treatise names A the "electromagnetic momentum"; Heaviside and Hertz soon banish potentials as unphysical scaffolding.

1959

— Aharonov and Bohm predict a measurable phase from A in field-free regions; Chambers reports the effect within a year.

1986

— Tonomura's shielded-toroid holography experiment closes every loophole: the potential's loop integral, not the local field, moves the fringes.

The physical picture

Electrostatics earned its potential from a curl: $\nabla\times\vec E=0$ meant closed-loop line integrals of $\vec E$ vanished, so a path-independent scalar $V$ existed and $\vec E=-\nabla V$ . Magnetostatics cannot copy that move — $\nabla\times\vec B=\mu_0\vec J$ is emphatically nonzero where currents flow, so no single-valued magnetic "voltage" exists in current-carrying regions. What magnetism conserves is the other differential statement: $\nabla\cdot\vec B=0$ , no sources, no sinks, field lines that never end. And the vector-calculus chapter proved the matching converse pair: curl-free fields are gradients, divergence-free fields are curls. Electricity's zero buys a scalar potential; magnetism's zero buys a vector one. That asymmetry — promised an explanation back when the divergence article first flagged it — is now cashed: it was never a stylistic choice, but the shape of which conservation law each field offers. The best intuition for $\vec A$ is genealogical: $\vec A$ looks like its parent current. A wire carrying current up generates an $\vec A$ that points up, strongest near the wire, weakening with distance — exactly the portrait of $V$ near a charged line. The magnetic field then wraps around because taking a curl turns "parallel and fading with distance" into "circulating." If you can sketch the current, you can sketch $\vec A$ without solving anything; $\vec B$ is one mechanical curl away. Fluid dynamicists know the same object by another name — for divergence-free flow, the velocity field is the curl of a stream function, and $\vec A$ is magnetism's stream function.

Figure

A inherits the geometry of the current while B wraps around it — wire and solenoid portraits

The flux relation is what saves $\vec A$ from being pure bookkeeping. Walk any closed loop and total up the tangential $\vec A$ : Stokes' theorem converts that circulation into the flux of $\nabla\times\vec A=\vec B$ through the loop. So $\vec A$ stores flux information locally, along curves — and it keeps the records even where the field itself has left. Outside an ideal solenoid $\vec B$ is exactly zero, yet a loop drawn around the solenoid must report the trapped interior flux; something on the loop has to carry that information, and that something is $\vec A=\frac{\Phi_B}{2\pi s}\hat\phi$ . Classically this looks like an accounting curiosity. Quantum mechanically it is an experiment: electrons steered around opposite sides of a shielded flux tube interfere as if they felt what they never touched.

The definition and its immediate payoffs

The definition, and the identity that makes it worth having:

\vec B=\nabla\times\vec A \qquad (1)

where: $\vec A$ — the magnetic vector potential, units $\mathrm{T\,m}=\mathrm{Wb/m}=\mathrm{V\,s/m}$ ; equivalently kg·m/(s·C) — momentum per unit charge, a reading that will earn its keep below. Because $\nabla\cdot(\nabla\times\vec A)\equiv0$ for any smooth $\vec A$ , equation (1) makes $\nabla\cdot\vec B=0$ automatic; and because $\nabla\cdot\vec B=0$ is exactly the hypothesis of the E0 converse theorem (divergence-free fields are curls, on well-behaved domains), such an $\vec A$ is guaranteed to exist for every magnetic field there is. One of Maxwell's two magnetostatic equations is hereby retired — permanently absorbed into notation. The non-uniqueness must be stated before it can be postponed. Since the curl of a gradient vanishes identically,

\vec A\;\to\;\vec A+\nabla\lambda \quad\text{leaves}\quad \vec B\ \text{unchanged, for any smooth }\lambda(\vec r). \qquad (2)

where: $\lambda$ — an arbitrary scalar "gauge function." Infinitely many $\vec A$ 's describe each $\vec B$ ; the choice among them is called a gauge choice, and the standard magnetostatic pick, $\nabla\cdot\vec A=0$ (the Coulomb gauge), is the one this chapter's integral formula will quietly live in. What the freedom means, why the Coulomb gauge is always reachable, what freedom remains after it, and how it converts Ampère's law into three Poisson equations — all of that is the next article, which owns the gauge story. Here we need only the fact of (2) and one gauge-proof quantity:

\oint_{\partial S}\vec A\cdot\mathrm{d}\vec\ell =\int_S(\nabla\times\vec A)\cdot\mathrm{d}\vec a =\int_S\vec B\cdot\mathrm{d}\vec a =\Phi_B . \qquad (3)

where: $S$ — any surface spanning the loop $\partial S$ . Stokes' theorem, applied to the definition. Read it twice: once as meaning ( $\vec A$ 's circulation is the enclosed flux — the vector potential is a flux ledger), and once as invariance (add $\nabla\lambda$ and the circulation shifts by $\oint\nabla\lambda\cdot\mathrm{d}\vec\ell=0$ around any closed loop, so the ledger's entries are gauge-independent). Equation (3) is to $\vec A$ what " $V$ differences are work per charge" was to the scalar potential: the license to compute with a non-unique object. And the door to mechanics, stated here and spent in the quantum-facing sections: a charged particle in a magnetic field carries the Lagrangian $L=\tfrac12mv^2+q\vec v\cdot\vec A-qV$ , whose conjugate momentum is not $m\vec v$ but

\vec p=m\vec v+q\vec A . \qquad (4)

where: $\vec p$ — the canonical momentum, the object quantum mechanics promotes to $-i\hbar\nabla$ . Maxwell, working before vectors had their modern clothes, called $\vec A$ the electromagnetic momentum for exactly this reason: it is the momentum a unit charge holds on deposit in the field, the quantity whose sudden release you feel as an induced kick when a nearby current is switched off. That reading — $\vec A$ as momentum-per-charge — is the single best answer to "but what is it?"

Derivation AThe integral formula for A, certified by one curl

The goal is the workhorse formula of the article: for steady, localized currents,

\vec A(\vec r)=\frac{\mu_0}{4\pi}\int\frac{\vec J(\vec r')}{\mathfrak{r}}\,\mathrm{d}\tau', \qquad \vec A(\vec r)=\frac{\mu_0 I}{4\pi}\oint\frac{\mathrm{d}\vec\ell'}{\mathfrak{r}} \quad\text{(line dialect)},

with $\mathfrak{r}=|\vec r-\vec r'|$ — the exact structural copy of $V=\frac{1}{4\pi\varepsilon_0}\int\frac{\rho}{\mathfrak{r}}\mathrm{d}\tau'$ , one copy per Cartesian component, with $\rho/\varepsilon_0\to\mu_0J_i$ . The strategy is deliberately honest about logical order: rather than deriving this from a Poisson equation (that route needs the Coulomb gauge, owned next article), we take the formula as candidate and certify it directly — its curl must return the Biot–Savart field, which is the definition of being a correct vector potential. This also repairs the base text's garbled sign bookkeeping: done carefully, no ambiguous signs survive.

Step 1. Set up the curl. Differentiation at the field point passes through the source integral:

\nabla\times\vec A =\frac{\mu_0}{4\pi}\int\nabla\times\!\left[\frac{\vec J(\vec r')}{\mathfrak{r}}\right]\mathrm{d}\tau' .

Why this is legal: $\nabla$ differentiates with respect to $\vec r$ while the integration variable is $\vec r'$ ; for smooth localized sources the derivative and integral commute.

Step 2. Product rule with a frozen vector. Inside the integral, $\vec J(\vec r')$ is a constant as far as $\nabla$ is concerned. The E0 product rule $\nabla\times(f\vec c)=(\nabla f)\times\vec c$ for constant $\vec c$ gives

\nabla\times\!\left[\frac{\vec J(\vec r')}{\mathfrak{r}}\right] =\nabla\!\left(\frac{1}{\mathfrak{r}}\right)\times\vec J(\vec r') =-\frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^2}\times\vec J(\vec r') =\frac{\vec J(\vec r')\times\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^2},

using the standard kernel gradient $\nabla(1/\mathfrak{r})=-\hat{\boldsymbol{\mathfrak{r}}}/\mathfrak{r}^2$ and one clean anticommutation of the cross product — the two minus signs meet and cancel, no $(\pm1)$ hedging required.

Step 3. Recognize the result.

\nabla\times\vec A =\frac{\mu_0}{4\pi}\int\frac{\vec J(\vec r')\times\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^2}\,\mathrm{d}\tau' =\vec B_{\rm Biot\text{–}Savart}(\vec r)\;\;\checkmark

— the volume-dialect Biot–Savart law, verbatim. The candidate is certified: it is a vector potential for the magnetostatic field.

Step 4. Which gauge did we land in? Take the divergence. Using $\nabla\cdot(f\vec c)=\vec c\cdot\nabla f$ and the kernel's exchange symmetry $\nabla(1/\mathfrak{r})=-\nabla'(1/\mathfrak{r})$ :

\nabla\cdot\vec A =\frac{\mu_0}{4\pi}\int\vec J\cdot\nabla\frac{1}{\mathfrak{r}}\,\mathrm{d}\tau' =-\frac{\mu_0}{4\pi}\int\vec J\cdot\nabla'\frac{1}{\mathfrak{r}}\,\mathrm{d}\tau' =\frac{\mu_0}{4\pi}\int\frac{\nabla'\cdot\vec J}{\mathfrak{r}}\,\mathrm{d}\tau' =0 .

The third equality is integration by parts (the surface term dies because the current is localized); the last is the steady-current condition $\nabla'\cdot\vec J=0$ from the continuity article. The formula is not gauge-agnostic: it delivers specifically the divergence-free representative — the Coulomb-gauge $\vec A$ , a fact the next article will re-derive from the Poisson side and lean on heavily. Verified: dimensions — $\mu_0 J\,\mathrm{d}\tau/\mathfrak{r}\sim(\mathrm{T\,m/A})(\mathrm{A/m^2})(\mathrm{m^3})/\mathrm{m}=\mathrm{T\,m}$ ✓; matches Griffiths Eq. 5.65 and Jackson §5.4; the by-parts step is the same maneuver the E1 energy article used, recalled not reinvented.

The formula's fine print is the same as its electrostatic parent's. It presumes the current is localized — for an infinite wire or solenoid the integral diverges, exactly as $V$ of an infinite line charge did, and the cure is identical: compute differences, or pull the potential from the field equations with a reference point, accepting a logarithm. And it presumes steady currents twice over — once for Biot–Savart to be the right target, once in Step 4's divergence check.

Go deeper · Derivation B · 2 minThe solenoid's vector potential, inside and out

Derivation BThe solenoid's vector potential, inside and out

The goal is the complete Coulomb-gauge $\vec A$ of the ideal infinite solenoid ( $n$ turns per meter, current $I$ , radius $R$ ) — the single most instructive $\vec A$ in physics, because it is nonzero precisely where the field is zero:

\vec A(s)=\frac{\mu_0 nI\,s}{2}\,\hat\phi\;\;(s\le R), \qquad \vec A(s)=\frac{\mu_0 nI\,R^2}{2s}\,\hat\phi=\frac{\Phi_B}{2\pi s}\,\hat\phi\;\;(s\ge R).

The strategy showcases why $\vec A$ earns its keep: instead of integrating anything, we use symmetry plus the flux relation (3) — an "Ampère's law for $\vec A$ " in which flux plays the role of enclosed current.

Step 1. Symmetry pins the direction. $\vec A$ inherits the source geometry: the solenoid's current is azimuthal, so $\vec A=A(s)\,\hat\phi$ — circles around the axis, magnitude depending only on $s$ . (Formally: each current element contributes $\mathrm{d}\vec A\parallel\mathrm{d}\vec\ell'$ by the integral formula, and the axial/rotational symmetry kills all but the azimuthal average.)

Step 2. Circulate on a coaxial circle. For a circle of radius $s$ , equation (3) reads $A(s)\cdot2\pi s=\Phi_B(s)$ , the flux through the circle. Two regimes:

Step 3. Inside ( $s\le R$ ): the enclosed flux is $B\cdot\pi s^2$ with $B=\mu_0 nI$ from the Ampère's-law article, so

A(s)=\frac{\mu_0 nI\,s}{2} \quad\Longleftrightarrow\quad \vec A=\tfrac12\,\vec B\times\vec s ,

the linear-in- $s$ circulating profile — one standard representative of a uniform field (the "symmetric gauge," one member of the menagerie the next article will display).

Step 4. Outside ( $s\ge R$ ): the enclosed flux saturates at $\Phi_B=\mu_0 nI\cdot\pi R^2$ — the solenoid holds all its field inside — so

A(s)=\frac{\Phi_B}{2\pi s}:

nonzero, falling as $1/s$ , carrying the memory of the trapped flux through a region where $\vec B=0$ identically. Why no gauge trickery can remove the exterior $\vec A$ : any candidate $\vec A$ must satisfy (3), and the circulation around the solenoid is the gauge-invariant number $\Phi_B\neq0$ . You can reshape the exterior potential, but you cannot make it vanish on every loop — some tangential $\vec A$ must survive to carry the flux record. That immovable remainder is what the electron reads in the Aharonov–Bohm experiment. *Verified: curl check in cylindrical coordinates — $B_z=\frac{1}{s}\partial_s(sA_\phi)$ gives $\frac1s\partial_s(\mu_0nIs^2/2)=\mu_0nI$ inside ✓ and $\frac1s\partial_s(\Phi_B/2\pi)=0$ outside ✓; continuity at $s=R$ : both branches give $\mu_0nIR/2$ ✓; numbers — $n=1000$ turns/m, $I=100$ A, $R=2$ cm: $B=0.126$ T, $\Phi_B=1.58\times10^{-4}$ Wb, $A=6.3\times10^{-4}$ T·m at $s=1$ cm and $5.0\times10^{-4}$ T·m at $s=5$ cm, recomputed ✓.*

Figure

The solenoid payoff profile — A rises linearly inside, falls as one over s outside, while B drops dead at the wall

Limiting cases

case	result	meaning
far from any localized current	$\vec A=\frac{\mu_0}{4\pi}\frac{\vec m\times\hat r}{r^2}$	the dipole term leads — no monopole term exists; derived in the multipole article next
uniform field region	$\vec A=\tfrac12\vec B\times\vec r$ (one choice of many)	the symmetric gauge; alternatives differ by $\nabla\lambda$ — next article's menagerie
outside confined flux	$\vec A=\frac{\Phi_B}{2\pi s}\hat\phi$	$\vec B=0$ but $\vec A$ carries the flux record; Aharonov–Bohm territory
infinite line current	$\vec A=-\frac{\mu_0I}{2\pi}\ln(s/s_0)\,\hat z$	integral formula diverges; reference-point logarithm, like the line charge's $V$
time-dependent sources	$\vec A$ obeys a wave equation; $\vec E=-\nabla V-\partial_t\vec A$	statics is the frozen limit; retarded potentials owned in the radiation chapter

Common trapA is a real field with unmeasurable parts — keep the two facts separate

Two opposite mistakes are equally tempting. Mistake one: "only $\vec B$ is real; $\vec A$ is bookkeeping" — refuted by quantum mechanics, where the canonical momentum is $m\vec v+q\vec A$ and the Aharonov–Bohm phase $q\Phi_B/\hbar$ is measured in interferometers while the electron flies exclusively through $\vec B=0$ territory. Mistake two: "then $\vec A(\vec r)$ itself must be measurable" — refuted by gauge freedom, since $\vec A$ and $\vec A+\nabla\lambda$ describe identical physics, so no experiment can report "the value of $\vec A$ here." What is physical is exactly the gauge-invariant content: $\vec B$ , and closed-loop integrals $\oint\vec A\cdot\mathrm{d}\vec\ell=\Phi_B$ (the AB phase is one of these, multiplied by $q/\hbar$ ). A parallel trap: the integral formula $\frac{\mu_0}{4\pi}\int\vec J/\mathfrak{r}\,\mathrm{d}\tau'$ silently assumes localized steady currents and delivers specifically the Coulomb-gauge representative — for infinite wires it diverges and you must fall back on reference points, exactly as with the line charge's logarithmic potential.

Worked examples

1 · The solenoid ledger, with numbers (confidence builder)

A solenoid has $n=1000$ turns/m, $I=100$ A, radius $R=2.0$ cm. Find $A$ at $s=1.0$ cm and $s=5.0$ cm, and the flux both values report.

Step 1. Interior field and flux. $B=\mu_0 nI=4\pi\times10^{-7}\times10^5=0.126$ T; trapped flux $\Phi_B=B\pi R^2=0.126\times\pi\times(0.02)^2=1.58\times10^{-4}$ Wb.

Step 2. Inside point. $A(1\ \mathrm{cm})=\mu_0nIs/2=0.126\times0.01/2$ = $6.3\times10^{-4}$ T·m, azimuthal. Check via the ledger: circulation $=A\cdot2\pi s=3.95\times10^{-5}$ Wb $=B\cdot\pi s^2$ ✓ — the circle's own enclosed flux.

Step 3. Outside point. $A(5\ \mathrm{cm})=\Phi_B/2\pi s=1.58\times10^{-4}/(2\pi\times0.05)$ = $5.0\times10^{-4}$ T·m — barely smaller than the interior value, in a region where $\vec B$ is identically zero. Its circulation: $A\cdot2\pi s=1.58\times10^{-4}$ Wb $=\Phi_B$ ✓ — every exterior loop files the same flux report.

Step 4. The quantum reading. An electron encircling this solenoid would acquire AB phase $e\Phi_B/\hbar\approx2.4\times10^{11}$ rad — absurdly scrambled, which is why real AB experiments shrink the flux to a few quanta: the natural unit is $h/e=4.14\times10^{-15}$ Wb, and our bench solenoid traps $\sim4\times10^{10}$ of them. Tonomura's decisive 1986 experiment used micron-scale magnets holding single-digit quanta.

2 · The infinite wire's logarithmic A (exam level)

Find $\vec A$ of an infinite straight wire carrying $I=15$ A, and verify it reproduces the known field. How much does $A_z$ change from $s=1$ cm to $s=2$ cm?

Step 1. Direction and the divergence honesty. $\vec A\parallel\vec J$ : purely $\hat z$ , magnitude $A_z(s)$ . The integral formula applied naively gives $\frac{\mu_0I}{4\pi}\int_{-\infty}^{\infty}\mathrm{d}z'/\sqrt{s^2+z'^2}$ — logarithmically divergent, the same disease as the line charge's $V$ . Cure: demand the right curl instead, with a reference radius $s_0$ .

Step 2. Solve the curl condition. In cylindrical coordinates with $\vec A=A_z(s)\hat z$ , the curl has the single component $B_\phi=-\partial A_z/\partial s$ . Setting $-\partial_sA_z=\mu_0I/2\pi s$ (the wire's known field) and integrating:

\vec A(s)=-\frac{\mu_0 I}{2\pi}\ln\!\frac{s}{s_0}\,\hat z .

Positive (parallel to $I$ ) for $s<s_0$ , negative beyond — the offset is pure reference convention, the logarithm's differences are physics.

Step 3. Numbers. $\Delta A_z=A_z(1\ \mathrm{cm})-A_z(2\ \mathrm{cm})=\frac{\mu_0I}{2\pi}\ln2=2\times10^{-7}\times15\times0.693$ = $2.1\times10^{-6}$ T·m. Field check at $s=1.5$ cm: $B=\mu_0I/2\pi s=0.20$ mT ✓ recomputed from the derivative.

Step 4. The momentum reading. By equation (4), switching this current off transfers impulse $q\,\Delta\vec A$ to a nearby charge — a proton at 1 cm versus one at 2 cm differ by $e\Delta A_z=3.3\times10^{-25}$ kg·m/s of deposited field momentum: tiny, but this ledger entry is exactly what Faraday's induced EMF pays out, as the electrodynamics chapter will make quantitative.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.4 · Jackson, Classical Electrodynamics, 3rd ed., §5.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §6.3 · Zangwill, Modern Electrodynamics, §10.1–10.3

Provenance. Equations cross-checked against Griffiths Introduction to Electrodynamics 4th ed §5.4 (Eqs. 5.62–5.65, dipole 5.85), Jackson Classical Electrodynamics 3rd ed §5.4, and Zangwill Modern Electrodynamics ch. 10. A verifier should recompute: the double-curl identity reduction to $\nabla^2\mathbf{A}=-\mu_0\mathbf{J}$ ; the integration-by-parts proof that the integral solution is divergence-free for steady currents; recovery of Biot–Savart by taking the curl; and the Aharonov–Bohm phase $q\Phi_B/\hbar$ . The value $\mu_0=4\pi\times10^{-7}\,\mathrm{T{\cdot}m/A}$ (exact pre-2019 SI) is standard. (Scope note: the double-curl/Poisson reduction is stated here and owned by the next article; this article certifies the integral solution directly by curl. The base text's Biot–Savart recovery carried ambiguous sign placeholders — repaired here with the frozen-vector product rule, no sign ambiguity remaining. Worked numbers: solenoid $0.126$ T / $1.58\times10^{-4}$ Wb / $6.3\times10^{-4}$ and $5.0\times10^{-4}$ T·m; wire $\Delta A_z=2.1\times10^{-6}$ T·m — recomputed.)

Builds on: Divergence of B and Absence of Magnetic Monopoles · Curl of B · Ampere's Law · Electric Current and Current Density · Biot-Savart Law

Leads to: Gauge Freedom and the Coulomb Gauge · Magnetic Field of Steady Current Configurations · Lorentz Force Law

Why does the magnetic field get a vector potential when the electric field only needed a scalar?

Each field's potential is purchased with its conservation law, and the two fields conserve different things. $\vec E$ (in statics) has zero curl — closed-loop work vanishes — and a curl-free field is the gradient of a scalar: one function, three components recovered by differentiation. $\vec B$ has zero divergence — no sources or sinks — and a divergence-free field is the curl of a vector: you need three functions, because the object being encoded (a circulating field) has intrinsically more directional structure than a "downhill flow." The asymmetry traces straight back to the sources: charges exist, monopoles don't. In a region with genuinely no current you can locally use a magnetic scalar potential too, a trick the magnetized-matter chapter will exploit — but it fails on any loop encircling current, where the would-be potential comes back multivalued.

Is A physically real, or just a calculational trick?

The modern verdict has two clauses. The configuration $\vec A(\vec r)$ , point by point, is not measurable — gauge freedom means infinitely many $\vec A$ 's describe one world, so asking "what is $\vec A$ here?" is like asking a mountain's "true altitude" without agreeing on sea level. But the gauge-invariant content of $\vec A$ is as real as anything in physics: quantum interference measures $\oint\vec A\cdot\mathrm{d}\vec\ell$ directly (Aharonov–Bohm), the canonical momentum $m\vec v+q\vec A$ is what quantum mechanics quantizes, and no local function of $\vec B$ alone can reproduce these effects when the flux is confined away from the particle. Summary: real field, redundant description. Classical electromagnetism let you pretend only $\vec B$ existed; quantum mechanics revoked that option.

What are the units of A, and is there an intuition for them?

$[\vec A]=\mathrm{T\cdot m}=\mathrm{Wb/m}=\mathrm{V\cdot s/m}$ — and, most instructively, kg·m/(s·C): momentum per unit charge. That last reading is Maxwell's own ("electromagnetic momentum") and it is operational: by $\vec p=m\vec v+q\vec A$ , a charge sitting in a vector potential holds $q\vec A$ of momentum on deposit in the field. Switch the source current off and the deposit is paid out — the induced electric field $-\partial\vec A/\partial t$ delivers exactly the impulse $q\vec A_{\rm initial}$ . The V·s/m version is equally practical: it is why $\vec A$ 's time derivative is an electric field, the fact the induction chapter runs on.

How literally should I take the analogy between A and the electrostatic potential V?

Component by component, very literally — with three caveats. The Coulomb-gauge $A_x$ is computed from $\mu_0J_x$ exactly as $V$ is from $\rho/\varepsilon_0$ , so every electrostatic solving technique (superposition, symmetry, even separation of variables) transfers. Caveat one: it's three parallel problems, coupled only through the source geometry. Caveat two: the analogy runs through the Coulomb gauge — in other gauges $\vec A$ obeys different equations. Caveat three: the physical readings differ — $V$ is energy per charge, $\vec A$ is momentum per charge; equipotentials of $V$ guide statics intuition, while for $\vec A$ the useful geometric object is its circulation, the flux ledger. Where the analogy is weakest: boundary conditions at material surfaces, which for $\vec A$ involve surface currents rather than surface charges (the boundary-conditions article ahead).

If B is zero outside a solenoid, how can anything out there depend on A? Doesn't that violate locality?

It genuinely bothered people, Ehrenberg and Siday included, and the resolution sharpens the concept. Classically, nothing outside depends on $\vec A$ : the Lorentz force needs $\vec B$ , and no classical exterior experiment detects the trapped flux. Quantum mechanically, the phase of a charged wavefunction accumulates $\frac{q}{\hbar}\int\vec A\cdot\mathrm{d}\vec\ell$ along each path, and two paths enclosing the solenoid differ by $\frac{q}{\hbar}\oint\vec A\cdot\mathrm{d}\vec\ell=\frac{q\Phi_B}{\hbar}$ — gauge-invariant, hence measurable, and manifestly an effect of the enclosed flux rather than the local field. Locality survives in refined form: the interaction is local in the potentials ( $\vec A$ couples to the particle where the particle is), while the observable depends only on a global, gauge-invariant loop quantity. What died is the stronger claim that $\vec E$ and $\vec B$ alone, sampled along the trajectory, determine all physics.

When should I compute A first instead of going straight for B?

Three standard situations. First, when the current's geometry is simpler than the field's — $\vec A$ is parallel to $\vec J$ , so a planar or azimuthal current gives a one-component $\vec A$ where $\vec B$ has two; the multipole expansion (next-but-one article) is built on $\vec A$ for exactly this reason. Second, when flux is the quantity you actually want: inductance calculations and induction problems read $\Phi_B=\oint\vec A\cdot\mathrm{d}\vec\ell$ directly off the potential without ever constructing $\vec B$ . Third, in any Lagrangian, Hamiltonian, or quantum setting, where the formalism simply refuses to run on $\vec B$ : minimal coupling needs $\vec A$ itself. For a quick field at a symmetric point, though, Ampère's law or Biot–Savart remains the shorter road — $\vec A$ is a strategic instrument, not a universal shortcut.

The solenoid's exterior A falls as 1 over s. Doesn't that clash with A being like the potential of its source?

No — it is the analogy working correctly, just for an unusual source. The exterior $\vec A=\Phi_B/2\pi s\,\hat\phi$ is azimuthal because the source current is azimuthal, exactly as the parallel-rule predicts. The $1/s$ profile is forced by the flux ledger: every exterior circle must return the same $\Phi_B$ , so $A\cdot2\pi s$ must be constant. Compare the interior: flux grows as $s^2$ , so $A$ grows as $s$ . The two regimes stitch continuously at the wall (both give $\mu_0nIR/2$ ), and the kink in slope at $s=R$ is precisely where the surface current lives — a preview of the general rule (owned by the boundary-conditions article) that $\vec A$ stays continuous across current sheets while its normal derivative jumps.

Which gauge is the integral formula in, and did I choose it?

You chose it implicitly — a subtlety worth making explicit once. Transcribing the electrostatic Coulomb integral component by component produces one specific representative out of the infinite gauge family: taking its divergence and integrating by parts (using $\nabla'\cdot\vec J=0$ for steady currents) gives exactly zero, so the formula lands in the Coulomb gauge $\nabla\cdot\vec A=0$ automatically. Nothing is wrong with the other representatives — add any $\nabla\lambda$ and the curl still returns the correct $\vec B$ — but they are not what the integral computes. The next article turns this from an observation into a method: impose $\nabla\cdot\vec A=0$ deliberately, and Ampère's law collapses into the vector Poisson equation whose solution this integral is.