Ampere's Law

The last article ended with a theorem and a promissory note. Taking the curl of the Biot–Savart law proved $\nabla\times\vec B=\mu_0\vec J$ , and one application of Stokes' theorem minted the integral version: the circulation of $\vec B$ around any closed loop equals $\mu_0$ times the current threading it. What that article did not do is spend the result. This one does — because Ampère's law, spent well, is the single biggest labor-saving device in magnetostatics. Fields that cost a page of Biot–Savart integration (the solenoid's interior, worked ring-by-ring two articles back) or that Biot–Savart never delivered at all (the solenoid's off-axis uniformity, the toroid's confinement) fall out in three lines each. You have seen this movie before, with the electric cast. Gauss's law was exact everywhere but computationally useful only for the symmetric trio — sphere, cylinder, plane — where symmetry pinned $\vec E$ 's direction and dependence before any calculation. Ampère's law is the same bargain with circulation in place of flux: always true, and a one-line quotient precisely when symmetry lets you choose a loop on which $\vec B$ is constant and tangential. The magnetic trio is the solenoid, the toroid, and the infinite current sheet, and this article works all three in full, the way the Gauss applications article worked its trio. The skill being taught is the ritual: read the symmetry, declare what $\vec B$ can and cannot depend on, choose the Amperian loop that turns the circulation into $B$ times a length, count the threaded current honestly, divide. Master it here, because the same ritual returns twice more — for the auxiliary field $\vec H$ in magnetized matter, and for the Ampère–Maxwell law once time dependence breaks the steady-current fine print.

Key takeaways

Ampère's law: $\oint\vec B\cdot\mathrm{d}\vec\ell=\mu_0 I_{\rm enc}$ around any closed loop — the integral twin of $\nabla\times\vec B=\mu_0\vec J$ , exact for steady currents ( $\nabla\cdot\vec J=0$ makes $I_{\rm enc}$ surface-independent).
It is a computational shortcut only with symmetry: you must be able to argue, before calculating, that $\vec B$ is constant and tangential (or zero, or perpendicular) along the loop.
The trio, worked in full here: infinite solenoid $B=\mu_0 nI$ inside and exactly zero outside (uniform across the whole bore — the off-axis fact Biot–Savart couldn't reach); toroid $B=\mu_0 NI/2\pi s$ inside the winding and zero outside, for any cross-section shape; infinite sheet $B=\mu_0 K/2$ on each side, direction flipping across the sheet, independent of distance.
The thick wire is the calibration case: $B$ rises linearly as $\mu_0 I s/2\pi a^2$ inside, peaks at the surface, falls as $\mu_0 I/2\pi s$ outside — the interior profile the curl article used as its pointwise audit.
Counting $I_{\rm enc}$ is a right-hand-rule discipline: currents count once per net piercing of any surface spanning the loop, signed by the loop's orientation.
KEY IDEA: when symmetry pins the direction of $\vec B$ and the coordinate it depends on, an Amperian loop on which $\vec B$ is constant and tangential collapses $\oint\vec B\cdot\mathrm{d}\vec\ell=\mu_0 I_{\rm enc}$ to a one-line algebra problem — the magnetic mirror of the Gauss's-law ritual.

The core insight

Whenever the symmetry of a steady current distribution lets you guess the direction of $\vec B$ and the coordinate it depends on, choose an Amperian loop on which $\vec B$ is constant and tangential; then $\oint\vec B\cdot\mathrm{d}\vec\ell=\mu_0 I_{\rm enc}$ collapses to a one-line algebra problem for the field magnitude.

Where this lives today

The toroid calculation of this article, scaled up four orders of magnitude, is the beating heart of magnetic-confinement fusion. A tokamak's toroidal-field system is equation (3) with engineering attached: ITER's target of $5.3$ T at major radius $R_0=6.2$ m requires $NI=2\pi R_0 B/\mu_0\approx1.6\times10^{8}$ ampere-turns — delivered by 18 D-shaped superconducting coils carrying $\sim9$ MA-turns each, and the D-shape itself is a quiet tribute to Derivation C's cross-section-independence. The built-in $1/s$ gradient that made the inner wall of our toy toroid 50% stronger than the outer is a first-order design constraint: it sets the mechanical stress distribution on the coils and drives particle drifts the plasma physicists must tame. At the opposite end of the scale, Ampère's law is a measuring instrument. A Rogowski coil — a flexible solenoid bent into a closed loop and clipped around a conductor — literally performs $\oint\vec B\cdot\mathrm{d}\vec\ell$ in hardware: its output samples the tangential field all along the path, and by Ampère's law that integral reports the enclosed current regardless of the conductor's position, shape, or how many cables you looped around. Every clamp meter an electrician carries is Derivation A commercialized; power grids, particle accelerators, and plasma experiments (tokamaks again, diagnosing their own multi-mega-amp plasma current) run on the same three lines of vector calculus.

Origins · 1 minHans Christian Ørsted and how this was discovered

Hans Christian Ørsted

1777–1851 · Copenhagen

Ørsted was a professor of physics at Copenhagen and a card-carrying disciple of Kantian Naturphilosophie — he believed on philosophical grounds that nature's forces were one, and unlike most philosophers he checked. During a lecture demonstration in April 1820 he noticed a compass needle twitch as he switched a voltaic current on and off; where others might have filed the flicker away, Ørsted spent three months quantifying it and published that July in a four-page Latin pamphlet that detonated across Europe. The needle did not point toward the wire or along it — it settled sideways, tangent to circles around the wire: the first evidence that a field could circulate, the geometric germ of everything in this article. Within months Ampère had built the mathematics of interacting currents, and Biot and Savart the distance law; Ørsted's twitch became the circulation law within the decade of his death. He remained Denmark's scientific patriarch — founding its technical university, coining the word "electromagnetism," and giving the CGS unit of magnetic field its old name, the oersted.

History

1820

— April: Ørsted's compass twitches beside a current; July: his pamphlet announces the circulating magnetic action. By September, Ampère demonstrates wire–wire forces and begins the mathematics of currents.

1826

— Ampère's Mémoire delivers the complete theory of magnetic action between currents — the circulation law in embryonic, integral clothing.

1855

— Maxwell's "On Faraday's Lines of Force" recasts Ampère's results as field equations; the curl form and the loop form become two dialects of one law.

1861

— Maxwell adds the displacement current, repairing the law's failure for unsteady currents and predicting electromagnetic waves.

1968

— Results from the Soviet T-3 tokamak stun the fusion community; the toroid of Derivation C, at megampere scale, becomes the leading confinement architecture, culminating in ITER's 18-coil toroidal field system today.

The physical picture

Picture $\vec B$ as a current-counting instrument that you walk. Carry a compass needle once around any closed path, adding up the component of $\vec B$ along your direction of travel at every step. Ampère's law says the running total — the circulation — reads $\mu_0$ times the net current punching through your loop, and nothing else. A wire outside the loop bends the field along your path, strengthening it on some legs and weakening it on others, but its contributions cancel exactly by the time you return to the start: the $1/s$ falloff of a wire's field is precisely the decay rate at which "shorter, stronger arc near the wire" balances "longer, weaker arc far from it." The circulation meter is blind to bystanders. That blindness is the law's power: it converts a question about the field everywhere along a path into a question about sources through the path. But a meter reading is one number, and a field along a loop is infinitely many. One equation cannot determine a function — unless symmetry first collapses the function to a single unknown. That is the entire game. For the trio of this article, symmetry does exactly that:

Solenoidal / translational symmetry (the infinite solenoid): invariance along the axis and around it forces $\vec B$ to be axial with magnitude depending only on the distance from the axis. The natural loop is a rectangle with two legs parallel to the axis.

Axial symmetry (the toroid): invariance under rotation about the central axis forces $\vec B$ to be azimuthal — field lines are circles about the axis — with magnitude constant on each circle. The natural loop is one of those circles.

Planar symmetry (the infinite sheet): invariance under translations in the sheet's plane forces a uniform field parallel to the sheet and perpendicular to the current, antisymmetric across it. The natural loop is again a rectangle, straddling the sheet.

Figure

The magnetic trio with their Amperian loops — solenoid and rectangle, toroid and circle, sheet and straddling rectangle

Hold the electric mirror up once more, because the correspondence is a working tool, not a decoration. Gauss: flux through a closed surface counts enclosed charge; useful when a surface of constant $|\vec E|$ exists. Ampère: circulation around a closed curve counts threading current; useful when a curve of constant tangential $B$ exists. Sphere, cylinder, plane on one side; solenoid, toroid, sheet (and the wire, already banked) on the other. Even the failure modes rhyme — Gauss is useless for a charged cube, Ampère for a finite wire segment or a single loop, and for the same reason: the law stays true while the symmetry argument that extracts the field from under the integral sign collapses.

The law and the counting rules

The statement to be spent, with its fine print. Around any closed curve $C$ , traversed in a chosen direction,

\oint_C \vec B\cdot\mathrm{d}\vec\ell \;=\; \mu_0\, I_{\rm enc} \qquad (1)

where: $\mathrm{d}\vec\ell$ — line element along $C$ ; $\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m/A}$ ; and $I_{\rm enc}=\int_S\vec J\cdot\mathrm{d}\vec a$ — the net current through any surface $S$ spanning $C$ , its normal oriented by the right hand: curl your fingers along the travel direction and your thumb picks the positive side. Read the left side as the meter (walk the loop, accumulate tangential $\vec B$ ) and the right as what the meter is guaranteed to report. The guarantee has two clauses worth respecting. Any surface: a flat disk or a grotesquely bulging balloon spanning the same rim must give the same $I_{\rm enc}$ — true in magnetostatics because $\nabla\cdot\vec J=0$ (the continuity article's steady-current condition) forbids current lines from ending between two such surfaces. Net piercing: a wire that punches through your surface, loops around, and punches back counts zero, because its two crossings carry opposite signs. That single sentence is the toroid's confinement theorem in disguise, as Derivation C will show. For counting, the sign discipline in one line: with the loop direction chosen and the thumb-normal fixed, currents crossing with the normal enter $I_{\rm enc}$ positive, against it negative. Two wires threading a loop with $3\ \mathrm{A}$ up and $5\ \mathrm{A}$ down give $I_{\rm enc}=-2\ \mathrm{A}$ , and the circulation is $-2\mu_0\ \mathrm{A} \approx -2.5\times10^{-6}\ \mathrm{T\,m}$ — no geometry of the loop, no positions of the wires within it, ever enter. The ritual that converts (1) into fields — memorize it as a checklist, exactly parallel to the Gauss version:

step	Gauss's law	Ampère's law
read the symmetry	direction of $\vec E$ + its one variable	direction of $\vec B$ + its one variable
choose the probe	Gaussian surface of constant $E_\perp$	Amperian loop of constant tangential $B$
evaluate the geometry side	$E\times(\text{area})$	$B\times(\text{length})$
count the source	$Q_{\rm enc}/\varepsilon_0$ , signed	$\mu_0 I_{\rm enc}$ , signed by right hand
divide	$E=Q_{\rm enc}/\varepsilon_0 A$	$B=\mu_0 I_{\rm enc}/L$

The symmetry step is the load-bearing one, and it is an argument, not a guess: you must be able to say why every component you drop is forbidden — by a rotation, a translation, a reflection, or by $\nabla\cdot\vec B=0$ . The derivations below model the full argument each time.

Derivation AFrom the curl law to the loop law

The goal is to mint equation (1) honestly from the local law $\nabla\times\vec B=\mu_0\vec J$ , which the previous article proved from Biot–Savart — and to establish the surface-independence that makes the right-hand side well defined. The strategy is one application of Stokes' theorem plus one appeal to steady currents; the point of writing it out is to see exactly where each hypothesis enters, because both get revoked later (the curl law when currents become unsteady, and with it the unambiguous $I_{\rm enc}$ ).

Step 1. Take any closed curve $C$ and any smooth surface $S$ whose boundary is $C$ , with orientations matched by the right-hand rule. Integrate the curl law over $S$ :

\int_S(\nabla\times\vec B)\cdot\mathrm{d}\vec a \;=\; \mu_0\int_S \vec J\cdot\mathrm{d}\vec a .

Step 2. Apply Stokes' theorem (the E0 integral theorem: circulation is the surface integral of curl) to the left side:

\oint_C \vec B\cdot\mathrm{d}\vec\ell \;=\; \mu_0\int_S\vec J\cdot\mathrm{d}\vec a \;=\; \mu_0\,I_{\rm enc}.

Why this is not yet a complete statement: the left side knows only about $C$ , but the right side appears to depend on which $S$ we chose. If two spanning surfaces could disagree, the "law" would assign two values to one circulation.

Step 3. Close the gap with steady currents. Let $S_1$ and $S_2$ both span $C$ ; together (one orientation flipped) they bound a closed volume $V$ . The difference of their current integrals is the flux of $\vec J$ out of $V$ , and the divergence theorem converts it:

\int_{S_1}\vec J\cdot\mathrm{d}\vec a-\int_{S_2}\vec J\cdot\mathrm{d}\vec a \;=\;\oint_{\partial V}\vec J\cdot\mathrm{d}\vec a\;=\;\int_V \nabla\cdot\vec J\,\mathrm{d}\tau\;=\;0,

since $\nabla\cdot\vec J=0$ for steady currents — the continuity equation with $\partial\rho/\partial t=0$ . Every spanning surface counts the same current; $I_{\rm enc}$ belongs to the loop alone. Why the fine print matters: for a charging capacitor, $\nabla\cdot\vec J\neq0$ on the plates, Step 3 fails, and a surface dodging between the plates counts zero conduction current while a disk through the wire counts $I$ . The magnetostatic law is then genuinely ambiguous — the crack through which Maxwell's displacement current enters, in the electrodynamics chapter. *Verified: dimensions — $[\vec B][\ell]=\mathrm{T\,m}$ and $\mu_0 I = 4\pi\times10^{-7}\times[\mathrm{A}]\ \mathrm{T\,m}$ ✓; the thin-wire check — circle of radius $s$ around a wire, $B_\phi\cdot 2\pi s=\mu_0 I$ reproduces $B=\mu_0 I/2\pi s$ , the Biot–Savart result the law must contain ✓.*

That one-line wire check deserves a sentence of respect before the new material: the field that took an honest infinite integral in the Biot–Savart article is here the calibration of the method — symmetry says azimuthal circles, the circle of radius $s$ has circulation $2\pi s\,B$ , the quotient is $\mu_0 I/2\pi s$ . Everything the trio does below is that maneuver with better scenery.

The trio, worked in full

Go deeper · Derivation B · 2 minThe infinite solenoid, inside and out

Derivation BThe infinite solenoid, inside and out

The goal is the complete field of an ideal infinite solenoid — $n$ turns per unit length, each carrying $I$ , forming an effectively continuous azimuthal current sheet on a cylinder of radius $R$ : the result is $\vec B=\mu_0 nI\,\hat z$ everywhere inside (uniform across the bore, not just on the axis) and exactly zero everywhere outside. The strategy has two movements: first a symmetry interrogation that reduces $\vec B$ to a single unknown function $B_z(s)$ , then two rectangular Amperian loops that determine that function completely. This pays the catalog article's outstanding promise — it derived $\mu_0 nI$ on the axis by stacking rings and had to defer the off-axis claim to this ritual.

Step 1. Kill two components by symmetry. The source is invariant under translations along $\hat z$ and rotations about the axis, so $\vec B$ can depend only on $s$ . The radial component dies by flux: a coaxial Gaussian cylinder of radius $s$ and length $L$ has flux $2\pi s L\,B_s(s)$ through its wall (the end caps cancel by translation symmetry), and $\nabla\cdot\vec B=0$ forces it to vanish — $B_s=0$ . The azimuthal component dies by Ampère itself: a coaxial circular loop of radius $s$ threads zero axial current (the solenoid's current is purely azimuthal), so $2\pi s\,B_\phi=0$ — $B_\phi=0$ . What survives is $\vec B=B_z(s)\,\hat z$ : an axial field with unknown radial profile. Why the interrogation comes first: without it, "choose a loop where B is tangential" is circular reasoning. Every dropped component has a named executioner — that is what a symmetry argument is.

Step 2. The exterior field is a constant… Take a rectangle entirely outside the solenoid, legs of length $L$ parallel to the axis at radii $s_1<s_2$ , both $>R$ . The radial legs contribute nothing ( $B_s=0$ ), so the circulation is $[B_z(s_1)-B_z(s_2)]L$ . The loop threads no current at all. Hence $B_z(s_1)=B_z(s_2)$ : the exterior axial field is one constant, independent of distance.

Step 3. …and the constant is zero. A solenoid's field must vanish as $s\to\infty$ — at distances huge compared with $R$ the source looks like a stack of tiny dipole rings whose fields die off as $1/s^3$ (the catalog article's far-field result). A constant that vanishes at infinity is zero: $B_z(s>R)=0$ identically. The same rectangle argument run with both legs inside shows the interior field is likewise one constant, $B_{\rm in}$ — uniformity across the bore, proved at last.

Step 4. The straddling rectangle sets the constant. Now place the rectangle with one leg inside, one outside, legs of length $L$ . Circulation: $B_{\rm in}L - 0$ . Threaded current: the loop's surface is pierced by $nL$ turns, each carrying $I$ , all in the same sense — $I_{\rm enc}=nLI$ . Ampère's law:

B_{\rm in}L=\mu_0 nLI \;\;\Rightarrow\;\; \boxed{\,\vec B_{\rm in}=\mu_0 n I\,\hat z\,, \qquad \vec B_{\rm out}=0\,} \qquad (2)

Why the answer is radius-blind: $L$ cancels, and nothing in the argument ever used $R$ or the leg's position — every interior point sees the same field, every exterior point none. The solenoid is magnetism's parallel-plate capacitor: all field inside, none outside, uniform, edge effects exiled to infinity. *Verified: the axial value matches Griffiths Eq. 5.59 and the ring-stacking integral of the catalog article with $\theta_1,\theta_2\to$ the infinite limits ✓; numbers — $n=1000$ turns/m at $I=100$ A gives $B=4\pi\times10^{-7}\times10^5=0.126$ T, recomputed ✓; the wall jump $B_{\rm in}-B_{\rm out}=\mu_0 nI=\mu_0 K$ with $K=nI$ anticipates the boundary-conditions article exactly ✓.*

Go deeper · Derivation C · 2 minThe toroid — confinement for any cross-section

Derivation CThe toroid — confinement for any cross-section

The goal is the field of a toroidal coil — $N$ turns total, current $I$ , wound around a doughnut whose cross-section may be circular, square, or any shape whatever: the result is $\vec B=\dfrac{\mu_0 NI}{2\pi s}\,\hat\phi$ at distance $s$ from the central axis inside the winding, and zero everywhere outside. The strategy: axial symmetry pins the field to azimuthal circles; then a single circular Amperian loop reads off the answer, and the net-piercing rule delivers the celebrated confinement. The catalog article stated this field and deferred the circulation argument here.

Step 1. The field is azimuthal. The winding is symmetric under rotation about the central axis (for closely spaced turns, exactly so). Rotational invariance means $\vec B$ 's components in cylindrical coordinates $(B_s,B_\phi,B_z)$ depend only on $s$ and $z$ . Poloidal components ( $B_s$ , $B_z$ ) would require field lines that spiral through the doughnut's hole or close within a cross-sectional plane; running the coaxial-cylinder flux argument and the reflection symmetry through any plane containing the axis (the current distribution maps to itself while a poloidal $\vec B$ would flip) kills them. What survives: $\vec B=B_\phi(s,z)\,\hat\phi$ — field lines are circles around the central axis, riding inside the tube.

Step 2. One circle, one quotient. Choose the Amperian loop to be a field line: a horizontal circle of radius $s$ centered on the axis, passing through the interior of the winding. The circulation is $B_\phi\cdot2\pi s$ ( $\vec B$ tangential and, by symmetry, constant on the circle). The spanning disk is pierced once by each of the $N$ turns, all in the same sense: $I_{\rm enc}=NI$ . Therefore

B_\phi\,(2\pi s)=\mu_0 NI \;\;\Rightarrow\;\; \boxed{\,B=\frac{\mu_0 NI}{2\pi s}\quad\text{inside the winding}\,} \qquad (3)

where: $s$ — distance from the central axis, not from the tube's center. The field is stronger on the inner wall than the outer: a toroid with inner radius $20$ cm and outer $30$ cm at $NI=3000\ \mathrm{A\cdot turns}$ runs from $3.0$ mT down to $2.0$ mT — a built-in $1/s$ gradient that no winding trick removes.

Step 3. Outside, the count is zero — three ways. For a circle in the hole ( $s$ smaller than the inner radius): the spanning disk is pierced by no current at all. For a circle beyond the outer radius: every turn pierces the disk twice — once going up through the inner leg, once coming down through the outer — net zero. For a loop threaded through the doughnut like a key ring: the turns' current crosses in canceling pairs again, though here a small honest remainder survives — the winding advances azimuthally by one wire spacing per turn, so the coil carries a hidden net current $I$ around the torus, giving a weak $\mu_0 I/2\pi s$ field in the hole region, killed in practice by winding a compensating return loop. To the ideal-toroid order: $\vec B=0$ everywhere outside the tube. Why the shape of the cross-section never entered: Steps 1–2 used only rotational symmetry about the central axis and the count $NI$ . Circular, D-shaped (the tokamak choice), rectangular — equation (3) holds inside them all. That generality is pure Ampère; Biot–Savart would price each shape separately. *Verified: dimension and limit checks — far from the hole with a thin tube, set $s\approx R_0$ and $n=N/2\pi R_0$ : $B=\mu_0 nI$ , the straightened solenoid ✓ (bend a solenoid into a circle and (3) is what becomes of $\mu_0 nI$ ); numbers at $s=0.25$ m, $NI=3000$ : $B=4\pi\times10^{-7}\times3000/(2\pi\times0.25)=2.4$ mT, recomputed ✓.*

Go deeper · Derivation D · 2 minThe infinite current sheet

Derivation DThe infinite current sheet

The goal is the field of an infinite plane sheet in the $xy$ -plane carrying uniform surface current $\vec K=K\,\hat x$ (amps per meter of transverse width): the result is $\vec B=\mp\mu_0 K/2\,\hat y$ for $z\gtrless0$ — uniform, parallel to the sheet, perpendicular to the current, direction flipping across the sheet, and independent of distance, the magnetic twin of the charged plane's $\sigma/2\varepsilon_0$ . The strategy: planar symmetry plus a wire-pairing argument pins the direction; a straddling rectangle does the rest.

Step 1. Direction by dissection. Slice the sheet into infinite straight wires along $\hat x$ . A field point above the sheet sees each wire's azimuthal field; pair the wire at $+y_0$ with its mirror at $-y_0$ . Their field contributions tilt oppositely, and the vertical ( $\hat z$ ) parts cancel while the horizontal parts add — along $-\hat y$ above the sheet (right-hand rule around $\hat x$ -directed current), $+\hat y$ below. Translation invariance in $x$ and $y$ forbids any dependence on $x,y$ ; the pairing kills $B_z$ ; and $B_x$ vanishes because every source wire's field is perpendicular to the wire's own direction. Conclusion: $\vec B=B(z)\,(\mp\hat y)$ , antisymmetric, with one unknown magnitude $B(z)$ .

Step 2. The straddling rectangle. Take a rectangle in the $yz$ -plane (perpendicular to the current), legs of length $\ell$ parallel to $\hat y$ at heights $\pm z$ , connected by verticals. The vertical legs contribute nothing ( $\vec B$ has no $z$ -component); the two horizontal legs each run with the local field when traversed in the right-handed sense, contributing $B(z)\,\ell$ apiece:

\oint\vec B\cdot\mathrm{d}\vec\ell = 2B(z)\,\ell, \qquad I_{\rm enc}=K\ell .

Step 3. The quotient.

2B(z)\,\ell=\mu_0 K\ell \;\;\Rightarrow\;\; \boxed{\,\vec B= \begin{cases} -\dfrac{\mu_0 K}{2}\,\hat y, & z>0\\[2mm] +\dfrac{\mu_0 K}{2}\,\hat y, & z<0 \end{cases}} \qquad (4)

where: the magnitude $\mu_0K/2$ carries no $z$ — the same geometric conspiracy as the electrostatic plane: moving away weakens each wire's contribution but flattens more of the sheet into effectiveness, and for a $1/s$ elementary field the trade is exactly even. At $K=500\ \mathrm{A/m}$ the field is $\mu_0\times250=3.1\times10^{-4}$ T on each side, whether measured at a millimeter or a kilometer. Why this sheet is secretly the solenoid: roll the plane into a cylinder and the two results merge — inside the tube the "two sides" of the rolled sheet reinforce ( $\mu_0K/2+\mu_0K/2=\mu_0K=\mu_0 nI$ ), outside they cancel. The uniform interior and dead exterior of Derivation B are equation (4) bent into a circle. And the total jump across the sheet, $\Delta B_\parallel = \mu_0K$ , is the universal boundary condition — proved in general two articles ahead. *Verified: direction spot-check with the right-hand rule on a single wire ✓; the jump $\mu_0K$ matches the solenoid wall jump of Derivation B with $K=nI$ ✓; dimensions — $[K]=\mathrm{A/m}$ , so $\mu_0K$ carries tesla ✓.*

Limiting cases

case	result	meaning
steady currents, high symmetry	one-line quotient for $B$	the trio + wire: the law as production tool
steady currents, no usable symmetry	law true, field unobtainable from it	finite wire, single loop, Helmholtz pair — back to Biot–Savart
toroid with $s\approx R_0$ , thin tube	$B\to\mu_0 nI$	toroid degenerates to a bent solenoid
solenoid of finite length	$\mu_0 nI$ only deep inside	end field $\tfrac12\mu_0 nI$ ; the catalog article's $\theta_1,\theta_2$ formula takes over
unsteady currents	$\oint\vec B\cdot\mathrm{d}\vec\ell=\mu_0 I_{\rm enc}+\mu_0\varepsilon_0\,\mathrm{d}\Phi_E/\mathrm{d}t$	displacement current restores surface-independence — Ampère–Maxwell, owned in the electrodynamics chapter
inside magnetized matter	$\oint\vec H\cdot\mathrm{d}\vec\ell=I_{\rm free,enc}$	the same ritual with bound currents hidden in $\vec H$ — owned in the next chapter

Common trapAmpère's law is always true, so I can always use it to find B

The law holds for every loop around every steady current distribution — and for almost all of them it is useless for finding $\vec B$ , because usefulness requires what truth does not: a loop along which symmetry forces $\vec B$ to be constant-and-tangential or provably zero. For a finite wire segment, a square loop, or a pair of coils, no such loop exists; writing $B\cdot2\pi s=\mu_0 I_{\rm enc}$ there produces a number that is not the field (at best it is an average of the tangential component). The companion trap is sloppier counting: $I_{\rm enc}$ is the net, signed current through a surface spanning your loop — a wire that pierces the surface twice in opposite senses counts zero, which is exactly why the toroid confines and why "the solenoid's current" never threads an exterior circle. If you cannot name the symmetry that pins $\vec B$ , put Ampère down and pick up Biot–Savart.

Worked examples

1 · A busbar in cross-section (confidence builder)

A solid cylindrical busbar of radius $a=2.0$ mm carries $I=40$ A, uniformly distributed. Find $\vec B$ at $s=0.5$ mm, at the surface, and at $4.0$ mm.

Step 1. Symmetry and loop. Uniform axial $\vec J$ with cylindrical symmetry: the wire interrogation applies verbatim, $\vec B=B(s)\hat\phi$ , and the loop is a concentric circle of radius $s$ . The current density is $J=I/\pi a^2=40/(\pi\times(2.0\times10^{-3})^2)=3.18\times10^{6}\ \mathrm{A/m^2}$ .

Step 2. Inside ( $s\le a$ ): count the enclosed fraction. $I_{\rm enc}=J\pi s^2=I\,s^2/a^2$ . Then $B\cdot2\pi s=\mu_0 I s^2/a^2$ , so $B=\mu_0 I s/2\pi a^2$ — rising linearly from zero at the center. At $s=0.5$ mm: $B=(4\pi\times10^{-7}\times40\times5\times10^{-4})/(2\pi\times4\times10^{-6})$ = 1.0 mT.

Step 3. The surface is the peak. At $s=a$ both branches give $B=\mu_0 I/2\pi a = (2\times10^{-7}\times40)/(2\times10^{-3})$ = 4.0 mT — the maximum anywhere.

Step 4. Outside ( $s\ge a$ ): the wire result. All $40$ A enclosed regardless of $s$ : $B=\mu_0 I/2\pi s$ . At $s=4.0$ mm: 2.0 mT, falling as $1/s$ thereafter — about $0.27$ mT at 3 cm, several times Earth's field, which is why current-carrying bus work deflects nearby compasses and sensors. The profile — zero at the center, linear rise, sharp peak at the surface, $1/s$ tail — is the magnetic image of the charged ball's field from the Gauss applications article, and Step 2's interior value $B=\mu_0 Js/2$ is precisely the expression the curl article differentiated to audit $\nabla\times\vec B=\mu_0\vec J$ pointwise.

Figure

Field profile of the example coaxial cable — linear rise, one-over-s fall, shell cancellation to zero

2 · Designing a coaxial cable (exam level)

A coax carries $I=15$ A along a solid inner conductor of radius $a=1.0$ mm and returns it through a shell of inner radius $b=4.0$ mm, outer radius $c=5.0$ mm (uniform densities). Map $B(s)$ everywhere and state the design payoff.

Step 1. One symmetry, four counts. Every region shares the cylindrical interrogation: $\vec B=B(s)\hat\phi$ , circular loops, $B=\mu_0 I_{\rm enc}(s)/2\pi s$ . Only the bookkeeping of $I_{\rm enc}$ changes across the four regions — this is the whole exam technique.

Step 2. Core, $s\le a$ . $I_{\rm enc}=I s^2/a^2$ , so $B=\mu_0 I s/2\pi a^2$ : at $s=0.5$ mm, $B=(2\times10^{-7}\times15\times0.5)/(10^{-3})^2\times10^{-3}$ — compute cleanly: $B=\mu_0 I s/2\pi a^2 = 2\times10^{-7}\times15\times(5\times10^{-4})/(10^{-6})$ = 1.5 mT, rising to 3.0 mT at the core surface.

Step 3. Gap, $a\le s\le b$ . Full current enclosed: $B=\mu_0 I/2\pi s = 3\times10^{-6}/s$ . At $s=2.5$ mm: 1.2 mT; at the shell's inner face ( $4.0$ mm): 0.75 mT.

Step 4. Shell, $b\le s\le c$ . The return current subtracts progressively: $I_{\rm enc}=I\left[1-\frac{s^2-b^2}{c^2-b^2}\right]$ . At $s=4.5$ mm the bracket gives $1-(20.25-16)/(25-16)=0.528$ , so $I_{\rm enc}=7.9$ A and $B=\mu_0\times7.9/(2\pi\times4.5\times10^{-3})$ = 0.35 mT — already halved.

Step 5. Outside, $s\ge c$ . $I_{\rm enc}=15-15=0$ : $\vec B=0$ exactly. The cable is magnetically invisible — the design payoff. Equal-and-opposite coaxial currents confine the entire field to the interior, which is why coax feeds instruments without disturbing them and why the electric twin of this problem (the Gauss applications article's coax, all $\vec E$ between the conductors) built the same cable from the other field's side.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.3 · Jackson, Classical Electrodynamics, 3rd ed., §5.3–5.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6 · Zangwill, Modern Electrodynamics, §10.2

Provenance. All equations are standard textbook results. A verifier should: confirm $\oint\mathbf{B}\cdot\mathrm{d}\boldsymbol{\ell}=\mu_0 I_{\mathrm{enc}}$ as the Stokes-theorem integral of $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$ (Griffiths §5.3); check the wire field $B=\mu_0 I/2\pi s$ (Griffiths Eq. 5.39), solenoid $B=\mu_0 nI$ (Eq. 5.59), toroid $B=\mu_0 NI/2\pi r$ , and current-sheet $B=\mu_0 K/2$ ; verify the fat-wire profile is continuous at $s=a$ . The constant $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ is the conventional (pre-2019) defined value, accurate to within $\sim10^{-10}$ post-redefinition. Cross-check Jackson §5.3–5.4, Purcell Ch. 6, Zangwill §10.2. Worked numbers in this article (busbar 1.0/4.0/2.0 mT; coax 1.5/1.2/0.35/0 mT with $I_{\rm enc}=7.9$ A at 4.5 mm; toroid 3.0/2.4/2.0 mT; ITER $1.6\times10^8$ A-turns) recomputed independently.

Builds on: Curl of B · Biot-Savart Law · Electric Current and Current Density · Continuity Equation

Leads to: Magnetic Field of Steady Current Configurations · Magnetic Vector Potential · Gauge Freedom and the Coulomb Gauge

How do I actually decide between Biot–Savart and Ampère's law on a new problem?

Run the symmetry interrogation first, and be strict about it. Ask: can I name transformations (rotations, translations, reflections) that leave the source invariant and thereby pin both the direction of $\vec B$ and the single coordinate it depends on? If yes — infinite (or effectively infinite) wires, slabs, sheets, solenoids, toroids — Ampère's law delivers the field in three lines. If no — finite segments, single loops, coil pairs, anything with ends — the law remains true but underdetermined: one scalar equation cannot recover a field that varies along the loop. Then Biot–Savart is the production tool. "Effectively infinite" is a real category: a wire's field at 1 mm distance doesn't care that the wire ends 30 cm away, so Ampère's answer is excellent there even though the exact geometry is finite.

Is the field outside a real solenoid exactly zero, or approximately?

Three separate corrections, worth keeping distinct. First, finite length: a real solenoid leaks at its ends, and the ideal result only holds deep inside (at the end face, the axial field is already down to half). Second, the helical pitch: real windings advance along the axis, so a solenoid carrying $I$ also transports a net axial current $I$ — outside, that produces the weak azimuthal field $\mu_0I/2\pi s$ of a straight wire (at 100 A and half a meter, about $40\ \mu\mathrm{T}$ — comparable to Earth's field), which is why precision magnets add a compensating return winding. Third, discreteness: near the wall, at distances comparable to the turn spacing, the field ripples with the individual wires. The ideal-solenoid statement is exact only for the idealization — an infinite, continuous, purely azimuthal current sheet.

The toroid's enclosed count NI seems to ignore where the turns are. Why does the winding density not matter?

Because the circulation integral only asks whether a current pierces the spanning surface, never where. Thirty turns bunched on the left and thirty spread on the right both contribute $60I$ to a loop threading the tube. What non-uniform winding breaks is the symmetry premise: if the turn density varies with azimuth, the field is no longer constant along the circular loop, and while $\oint\vec B\cdot\mathrm{d}\vec\ell=\mu_0NI$ still holds, you can no longer pull $B$ out of the integral — you learn the average tangential field, not the field. Uniform winding is what upgrades the meter reading into a pointwise answer.

Why does the current sheet's field not weaken with distance? That seems unphysical.

It is the same geometric conspiracy as the electrostatic infinite plane. Decompose the sheet into wires: each wire's field falls as $1/s$ , but as you retreat, the patch of sheet that contributes effectively (the wires within roughly 45 degrees of overhead) grows in proportion. For a $1/s$ elementary field those two effects cancel exactly — retreat doubles the distance to each wire but doubles the recruited width. Physically, no real sheet is infinite: the result holds at distances small compared with the sheet's extent, exactly as the capacitor-plate result did. The deeper structural point survives idealization: crossing the sheet, $B_\parallel$ jumps by $\mu_0K$ no matter what — the local boundary condition owned two articles ahead.

When I compute I_enc, does the shape of the spanning surface ever matter?

For steady currents, never — that is a theorem, not a convenience. Two surfaces spanning the same loop differ by a closed surface, and the divergence theorem plus $\nabla\cdot\vec J=0$ forces equal counts through both. This licenses the standard trick of choosing whatever surface makes counting easiest (a flat disk for a toroid, a surface hugging the windings for a solenoid). The moment currents become unsteady — a capacitor charging, a wire with accumulating charge — the counts genuinely disagree, and that disagreement is not a technicality to wave away: it is the experimental crack in magnetostatics through which Maxwell inserted the displacement current. The ambiguity you learn to check here becomes the discovery there.

Ampère's law says outside currents contribute nothing. But an outside wire clearly changes B on my loop — how do both statements hold?

Distinguish the field from its circulation. An outside wire absolutely changes $\vec B$ at every point of your loop — Ampère's law never denies it. What the law asserts is that the closed-loop sum of those changes is zero: the outside wire strengthens the tangential field along the near arc and weakens (or reverses) it along the far arc, and the $1/s$ falloff makes the trade exact. This is also the practical warning hiding in the symmetric calculations: when you write $B\cdot2\pi s=\mu_0I_{\rm enc}$ for a wire, you are using symmetry to claim the wire's own field is uniform on the circle. Add a second wire nearby and the circulation equation stays true while the uniformity claim — and with it your ability to solve for $B$ — quietly dies.

Is the fat wire's linear interior rise related to the charged sphere's linear interior field?

Directly — they are the same bookkeeping in different clothes. Inside a uniform ball of charge, $Q_{\rm enc}\propto r^3$ while the Gaussian area grows as $r^2$ , leaving $E\propto r$ . Inside a uniform column of current, $I_{\rm enc}\propto s^2$ while the Amperian circumference grows as $s$ , leaving $B\propto s$ . In both, the field peaks at the surface and decays outside ( $1/r^2$ electric, $1/s$ magnetic — one power softer because the source is a line-like column rather than a point-like ball). The interior formula $B=\mu_0Js/2$ is also the expression whose curl the previous article computed to verify $\nabla\times\vec B=\mu_0\vec J$ pointwise — the integral and differential laws auditing each other.

What exactly breaks when currents are not steady, and how bad is it?

Quantitatively bad, not cosmetically. The derivation of surface-independence spends $\nabla\cdot\vec J=0$ ; with charge piling up anywhere ( $\partial\rho/\partial t\neq0$ ), two spanning surfaces of one loop can count completely different currents — for a charging capacitor, $I$ through a disk crossing the wire, exactly zero through a surface ballooning between the plates. The law as written then assigns two values to one circulation, i.e., it is no longer a law. The repair adds $\mu_0\varepsilon_0\,\mathrm{d}\Phi_E/\mathrm{d}t$ — Maxwell's displacement current — which restores surface-independence exactly because $\nabla\cdot\vec J=-\partial\rho/\partial t$ is compensated by the changing electric flux. For slowly varying currents (quasistatics: circuit frequencies, most lab magnets) the correction is negligible and this article's results stand as excellent approximations.