Ampère's Circuital Law

Within a week of hearing what Ørsted's compass had done, André-Marie Ampère stood before the Paris Academy with the beginnings of a theory — and he did not stop for six years, until magnetism had a complete mathematics. Buried in his 1826 masterwork is an idea of startling economy: instead of adding up a wire's field slice by slice, walk a closed loop around the current and keep a running total of the field along your path. That total, Ampère's circuital law says, doesn't depend on the loop's shape, or its size, or where the wire sits inside it. It depends on exactly one thing: how much current threads through. You have met this trade before. Gauss's Law took the labour of Coulomb-summing and replaced it, whenever symmetry allowed, with a two-line argument about flux through a closed surface. Ampère's law is the same bargain struck with magnetism: circulation around a closed loop in place of flux through a closed surface, threaded current in place of enclosed charge. And it pays the same way — results that cost the Biot–Savart recipe an infinite add-up fall out in two lines of algebra. The straight wire's field, stated on trust in Fields of Straight Wires & Circular Loops, is about to be derived before your eyes. The field inside a thick cable — new territory, and a JEE favourite — comes almost free. For CBSE, ISC, JEE and NEET this law is a statement-and-applications topic: know what the circulation means, count threaded current correctly (signs included), and run the symmetry argument for wires and — next module — solenoids. This article builds exactly those skills, plus the judgement the law demands: knowing when its magic works, and when it is true but useless.

Key takeaways

Ampère's circuital law: walk any closed loop and total up (field component along your step) × (step length); the grand total is $\mu_0$ times the current threading the loop: $\sum B_\parallel\,\Delta l = \mu_0 I_{\text{enc}}$ .
The total is blind to loop shape, loop size, and where inside the loop the currents sit — and currents outside the loop contribute exactly zero to it.
Solver mode needs symmetry: on a circle centred on a straight wire, $B$ is the same everywhere and tangent, so $B \times 2\pi r = \mu_0 I$ gives $B = \dfrac{\mu_0 I}{2\pi r}$ in one line.
Inside a thick wire of radius $R$ : only the fraction $r^2/R^2$ of the current is enclosed, so $B = \dfrac{\mu_0 I r}{2\pi R^2}$ — the field rises linearly from zero at the axis, peaks at the surface, then falls as $1/r$ .
Threaded current is a signed, net count: 5 A up and 2 A down through the same loop is $I_{\text{enc}} = 3$ A.
Key idea: circulation counts threaded current — always true for steady currents, and a two-line field solver whenever symmetry makes $B$ constant along the loop.

The core insight

Ampère's circuital law states that the sum of $B_\parallel\,\Delta l$ around any closed loop equals $\mu_0$ times the net current threading it — a universal current-counter that becomes a two-line field solver whenever symmetry pins $B$ constant and tangent along the loop.

Where this lives today

Every electrician's clamp meter is Ampère's law cast in plastic. Clip the jaws around a live cable — no contact, no circuit-breaking — and the instrument reads the current inside: the sensor ring is an Amperian loop, sampling the field around its circumference and reporting the total, which by the circulation law is $\mu_0 I_{\text{enc}}$ wherever the cable sits in the jaws. Clamp two wires carrying equal and opposite currents and it reads zero — the signed net count, live on a display. The laboratory version, the Rogowski coil, wraps the same theorem around megampere pulses in fusion machines and lightning research. The law's grandest deployment is the tokamak: fusion machines confine hundred-million-degree plasma on closed field-line loops, and the field budget — ampere-turns of coil per tesla around the ring — is circulation-law bookkeeping at industrial scale (ITER runs on roughly $10^8$ ampere-turns). In this course, the next stop is immediate: Solenoids & Toroids aims the two-line ritual at coil geometries, and the E4 problem gym drills the multi-region cable and cavity setups (Ampère Setups) that JEE-Advanced saves for its hardest magnetism slots.

The physical picture

Here is the law as something you do. Choose any closed path in space — a circle, a rectangle, a rubber-band squiggle — and walk it. At every step, note the component of the magnetic field pointing along your direction of travel, multiply by the step length, and keep a running total. Steps where the field pushes with you count positive; steps where it fights you count negative; steps where it crosses your path sideways count zero. The grand total when you arrive back at the start is called the circulation of the field around your loop. Ampère's law is the statement that this walking-total is a current meter: circulation around the loop $= \mu_0 \times$ (net current threading the loop). The astonishing part is what the meter ignores. Walk a small circle around a wire or a huge lopsided loop around the same wire: same total. Let the wire sit dead-centre or hug one edge: same total. Put a second, outside wire nearby — it visibly bends the field at every point of your path — and the total does not move at all: its help on the near arc and hindrance on the far arc cancel exactly by the time the loop closes. The meter reads threaded current, all of it, and nothing else.

Figure

Walk the loop and total B-along-step — the total reads only the threaded current

This "meter mode" is a genre of exam question all by itself, and it needs no formulas — just careful counting. But the law's real power is solver mode, and it works on exactly the logic Gauss's Law taught: one equation can only find one unknown. The circulation total is a single number; the field along a general loop is a million different values. Only when symmetry guarantees that $B$ has the same magnitude everywhere on the loop, and runs along it, does the total collapse to $B \times (\text{loop length})$ — one unknown, one equation, solved. Around a straight wire, the field's own circular pattern hands you the perfect loop on a plate — the calculation we now do.

The law and the counting rules

The statement, in the walking notation:

\sum B_\parallel\,\Delta l = \mu_0\, I_{\text{enc}} \qquad (1)

where: the sum runs once around any closed loop, $B_\parallel$ is the field component along the walking direction at each step $\Delta l$ , $\mu_0 = 4\pi \times 10^{-7}$ T·m/A, and $I_{\text{enc}}$ — the enclosed (threaded) current — is the net current punching through any surface that the loop borders, like a soap film on a ring. (In calculus notation this sum is written $\oint \vec B \cdot d\vec l$ ; the physics is the walk you just pictured.) Read the two sides as instrument and reading: the left is the meter you build out of field measurements; the right is what the meter is guaranteed to display. The counting discipline on the right side decides most of the marks. Threading means piercing the film. A wire that passes through the loop counts; a wire that runs beside it, or loops over it without passing through, does not. The count is signed. Curl your right-hand fingers along your walking direction; your thumb defines the positive threading direction. Currents through the film along the thumb count $+$ , against it count $-$ . The count is net. Three wires through one loop, carrying 4 A up, 1 A down, 2 A up: $I_{\text{enc}} = 4 - 1 + 2 = 5$ A, and the circulation is $\mu_0 \times 5 = 6.3 \times 10^{-6}$ T·m regardless of the loop's shape or the wires' positions inside it. A wire that pierces the film twice — in and back out, as a hairpin would — contributes twice with opposite signs: net zero. That innocent-looking bookkeeping fact is the secret of magnetic shielding designs, and (next module) of why a toroid's field never leaks out. The parallel with Gauss's Law is worth making one row at a time, because you already own the ritual on the electric side:

step	Gauss's Law (electric)	Ampère's law (magnetic)
the probe	closed surface	closed loop
the total	flux through it	circulation around it
what it counts	enclosed charge $/\varepsilon_0$	threaded current $\times \mu_0$
blind to	outside charges, positions inside	outside currents, positions inside
solver mode needs	symmetry: same $E$ over the surface	symmetry: same $B$ along the loop
the jackpots	sphere, long wire, large sheet	straight wire, solenoid, toroid

Derivation AThe straight wire field in two lines

The goal is the long straight wire's field $B = \mu_0 I/2\pi r$ — the formula Fields of Straight Wires & Circular Loops asked you to hold on trust because the honest Biot–Savart add-up needs calculus. The strategy is Ampère's law in solver mode: argue from symmetry that the field is the same size everywhere on a centred circle and points along it, then let the law collapse to one multiplication. The thing to watch is how the symmetry argument comes first — without it, pulling $B$ out of the sum would be illegal, and that legality check is exactly what separates correct Ampère solutions from confident nonsense.

Step 1. State the symmetry. An infinite straight wire looks identical from every side and from every height: rotate around it or slide along it and nothing changes. So the field's magnitude can depend only on the distance $r$ from the wire — not on the angle around it, not on the position along it. And the field's direction, as the right-hand thumb rule and Ørsted's compass both testify, is tangent to circles centred on the wire.

Step 2. Choose the loop to match: a circle of radius $r$ , centred on the wire, in a plane perpendicular to it, walked in the field's own direction. On this loop, every step has $B_\parallel = B$ (the full field, same magnitude at every step). The circulation collapses:

\sum B_\parallel\,\Delta l = B \sum \Delta l = B \times 2\pi r.

Why this collapse is the whole game: the law is true for every loop, but only this loop turns "a million unknown field values" into "one unknown number times a known length". Symmetry testifies first; algebra follows.

Step 3. Count the threaded current: the wire pierces the circle's film once, carrying $I$ . So $I_{\text{enc}} = I$ , and Ampère's law reads

B \times 2\pi r = \mu_0 I \quad\Longrightarrow\quad B = \frac{\mu_0 I}{2\pi r} \qquad (2)

Step 4. Appreciate what just happened. The Biot–Savart route to (2) sums infinitely many slice contributions with a genuine integral. Ampère's route: one symmetry sentence, one multiplication, one division. Same physics, same answer — the two laws are consistent, and the circulation law is the labour-saving form whenever the symmetry cooperates. Verified: matches equation (1) of Fields of Straight Wires & Circular Loops exactly; numbers — $I = 6.0$ A at $r = 4.0$ cm gives $B = 2\times10^{-7} \times 6.0/0.040 = 3.0 \times 10^{-5}$ T, recomputed; direction (tangent circles, thumb rule) consistent with the symmetry premise.

Go deeper · Derivation B · 2 minInside a thick wire — the enclosed fraction

Derivation BInside a thick wire — the enclosed fraction

The goal is the field inside a solid cylindrical conductor of radius $R$ carrying total current $I$ spread uniformly over its cross-section — a real cable, not an idealized line — at distance $r < R$ from the axis. The strategy is identical to the previous derivation with one new ingredient: the Amperian circle now catches only part of the current, and the enclosed fraction is pure geometry. The result — a field that rises with distance inside the conductor — surprises everyone the first time, and JEE examiners know it.

Step 1. Symmetry, verbatim from before: circles of constant $B$ , tangent direction. Choose an Amperian circle of radius $r$ inside the conductor. Circulation $= B \times 2\pi r$ , as always.

Step 2. Count the enclosed current — the new step. The current is spread uniformly, so the share threading our circle is the share of cross-sectional area it encloses:

I_{\text{enc}} = I \times \frac{\pi r^2}{\pi R^2} = I\,\frac{r^2}{R^2}.

Why the fraction is area-based: uniform spreading means every square millimetre of cross-section carries the same current; our circle encloses $\pi r^2$ of the total $\pi R^2$ . A non-uniform cable would change this one line and nothing else — that flexibility is the method's quiet strength.

Step 3. Apply the law and solve:

B \times 2\pi r = \mu_0 I\,\frac{r^2}{R^2} \quad\Longrightarrow\quad B = \frac{\mu_0 I\,r}{2\pi R^2} \qquad (3)

Step 4. Read the full profile. Inside, (3) rises linearly: zero at the axis (a circle of zero radius threads no current), climbing to $\dfrac{\mu_0 I}{2\pi R}$ at the surface. Outside, the previous derivation applies with the whole current enclosed: $B = \dfrac{\mu_0 I}{2\pi r}$ , falling. The field peaks at the surface — climb, cusp, decay. Why the interior rises: moving outward inside the cable, the enclosed current grows like $r^2$ while the loop length grows only like $r$ — the source wins. Outside, the enclosed current is frozen at $I$ and only the loop keeps growing — the geometry wins. The surface is where the contest changes hands. *Verified: the two branches agree at $r = R$ (both give $\mu_0I/2\pi R$ ) — no jump ✓. Numbers for the worked example's cable ( $R = 2.0$ mm, $I = 10$ A): surface field $2\times10^{-7}\times10/0.002 = 1.0$ mT; at $r = 1.0$ mm, (3) gives $0.50$ mT; at $r = 4.0$ mm outside, $0.50$ mT — the halfway-in and double-out points read the same, a symmetry of the climb-and-decay profile worth noticing. All recomputed.*

Figure

Field profile of a thick wire — linear climb inside, peak at the surface, 1/r decay outside

Limiting cases

case	result	meaning
circle around a thin wire	$B = \mu_0 I/2\pi r$	the two-line classic — equation (2)
inside a thick wire	$B = \mu_0 I r/2\pi R^2$ , rising	enclosed fraction $r^2/R^2$
at the axis of the thick wire	$B = 0$	zero radius threads zero current
at the surface $r = R$	both branches give $\mu_0 I/2\pi R$	the peak; profile is continuous
loop threading no current	circulation $= 0$	fields present, totals cancel — not "B = 0"
equal currents up and down through one loop	circulation $= 0$	net count; the shielding principle
no symmetry (finite wire, single loop)	law true, field unobtainable from it	back to Biot–Savart slices
ideal solenoid (preview)	$B = \mu_0 n I$ inside	next module cashes it in three lines

Common trapZero circulation means zero field

It does not — it means zero net threaded current, and the difference is a reliable mark-killer. Take a loop beside (not around) a live wire: the wire's field exists at every point of that loop, bending compasses and deflecting beams; yet the circulation totals exactly zero, because the with-you and against-you steps cancel on the closed round trip. The same confusion runs the other way: students see a symmetric-looking situation, compute a circulation, and divide by the loop length even though $B$ is not constant along it — manufacturing a number that is the average tangential field, not the field. The two-part discipline: circulation ↔ threaded current is always an honest pairing; extracting $B$ itself additionally requires the symmetry sentence ("same magnitude everywhere on the loop, directed along it"). No sentence, no solve — use Biot–Savart instead.

Worked examples

Board level — reading the meter, then using it (CBSE/ISC pattern, 3 marks)

Three long parallel wires pass perpendicular to the plane of a closed loop C, carrying 4.0 A and 2.0 A into the page and 5.0 A out of the page. A fourth wire carrying 8.0 A lies outside the loop. (a) Find the circulation of the magnetic field around C, walked clockwise. (b) A circular Amperian loop of radius 4.0 cm is now centred on a single long straight wire carrying 6.0 A; find B on that loop. (c) State why the 8.0 A wire affects the field at points of C but not the circulation.

Step 1. Part (a): fix signs. Walking clockwise (viewed from the front), the right-hand thumb points into the page — so into-page currents count positive. $I_{\text{enc}} = +4.0 + 2.0 - 5.0 = +1.0$ A. The outside wire never pierces the loop's film: it contributes nothing. Circulation $= \mu_0 I_{\text{enc}} = 4\pi\times10^{-7} \times 1.0 = \boxed{1.26 \times 10^{-6}\ \mathrm{T\,m}}$ .

Step 2. Part (b): symmetry sentence first — "by symmetry, B has the same magnitude at every point of the centred circle and is directed along it" — then the collapse: $B \times 2\pi(0.040) = \mu_0 (6.0)$ , so $B = \dfrac{2\times10^{-7} \times 6.0}{0.040} = \boxed{3.0 \times 10^{-5}\ \mathrm{T}}$ .

Step 3. Part (c), the sentence the mark scheme wants: "An outside current alters $\vec B$ at individual points of C, but around the closed loop its along-the-path contributions cancel exactly; the circulation counts only currents threading the loop."

Step 4. Sanity habit: a circulation is in tesla·metre and is typically $\mu_0 \times$ (a few amperes) $\sim 10^{-6}$ T·m; a field near a bench wire is $\sim 10^{-5}$ T. If your two answers' sizes swap, the formulas got crossed.

Competitive level — the cable's interior profile (JEE-Main pattern, ~3 minutes)

A solid cylindrical conductor of radius $R = 2.0$ mm carries $I = 10$ A distributed uniformly over its cross-section. Find the magnetic field (a) at $r = 1.0$ mm from the axis, (b) at the surface, (c) at $r = 4.0$ mm. (d) At what two distances is the field exactly half its maximum value?

Step 1. Maximum first — it anchors everything. The peak sits at the surface: $B_{\max} = \dfrac{\mu_0 I}{2\pi R} = \dfrac{2\times10^{-7} \times 10}{2.0\times10^{-3}} = \boxed{1.0\ \mathrm{mT}}$ .

Step 2. Part (a), inside, equation (3) — or faster, the linear read: at $r = R/2$ the interior field is half the surface value: $B = \boxed{0.50\ \mathrm{mT}}$ . (Check against (3): $2\times10^{-7} \times 10 \times 10^{-3}/(4.0\times10^{-6}) = 5.0\times10^{-4}$ T ✓.)

Step 3. Part (c), outside, full current enclosed: $B = \dfrac{2\times10^{-7} \times 10}{4.0\times10^{-3}} = \boxed{0.50\ \mathrm{mT}}$ — the same as (a). Not a coincidence: the linear climb reaches half-peak at $R/2$ , and the $1/r$ decay returns to half-peak at $2R$ .

Step 4. Part (d) is now already answered, and stating the general rule earns the concept mark: half-maximum occurs at $\boxed{r = R/2 \text{ and } r = 2R}$ — one point inside, one outside, for any uniform cable. Sketch the climb-cusp-decay profile and mark the symmetric pair; setters ask this exact pair because the unprepared re-derive both points from scratch. Timing note: three minutes, of which the physics is thirty seconds — peak at surface, halves at $R/2$ and $2R$ — and the rest is careful powers of ten (mm → m twice). If both your inside and outside answers don't come out equal at those mirror points, a conversion slipped. MCQ variant to expect: "ratio of field at $R/2$ to field at $2R$ " — answer 1, no calculator.

Exam traps

Dividing by the loop length without symmetry. $B \times (\text{length}) = \mu_0 I_{\text{enc}}$ is legal only after the symmetry sentence. For a square loop around a wire, or any loop around a finite wire, the collapse is illegal — the law still holds, but it cannot hand you $B$ .

Unsigned counting. Threaded currents carry signs set by your walking direction (right-hand rule). "Total current through the loop" means the net: 5 A up with 2 A down is 3 A, not 7 A.

Outside currents in the count. Only currents piercing the loop's film enter $I_{\text{enc}}$ . A wire beside the loop contributes zero to the circulation — however strongly it bends the local field.

Zero circulation read as zero field. A loop threading no net current has zero total; the field on it is generally nonzero everywhere. The converse error costs the same mark in reverse.

Inside-the-cable amnesia. Using $\mu_0 I/2\pi r$ at $r < R$ (with the full current) is the classic thick-wire blunder — the field would wrongly blow up toward the axis. Inside, the enclosed fraction $r^2/R^2$ tames it to a linear rise from zero.

Perpendicular-distance and unit slips. $r$ is measured from the wire's axis; cable problems love mm. Convert once, loudly, at the start.

Steady currents only. The law as stated holds for steady (unchanging) currents. The honest extension to changing currents — Maxwell's displacement current — is the crown of this track's final chapter; if a question mentions charging capacitors and circulation, that is where it lives.

Boards and constants. CBSE/ISC state-and-apply the law (wire, and solenoid next module); JEE-Adv adds interior-profile and multi-region problems; NEET keeps to the standard cases. IB, IGCSE, AP-2 and A-Level do not examine Ampère's law — their candidates need only the results it certifies. Constants: $\mu_0 = 4\pi\times10^{-7}$ T·m/A, $\mu_0/2\pi = 2\times10^{-7}$ .

Origins · 1 minAndré-Marie Ampère and how this was discovered

André-Marie Ampère

1775–1836 · Lyon and Paris

Ampère heard of Ørsted's discovery at the Academy on 4 September 1820 and read his own first paper on it fourteen days later — the start of a six-year sprint ending in the 1826 "Mémoire on the mathematical theory of electrodynamic phenomena", the book Maxwell called "perfect in form, and unassailable in accuracy" and its author "the Newton of electricity". (The unit on every ammeter, met back in Electric Current & Charge Flow, honours the same man; this article is his theorem, the deeper legacy.) Self-taught from his father's library in Lyon — a father guillotined in the Revolution when Ampère was seventeen — he gave physics the current-current force law, the circulation law bearing his name, and the audacious hypothesis that all magnetism, bar magnets included, is currents in disguise: molecular loops, guessed a century before atomic physics proved him right. His chosen epitaph: Tandem felix — happy at last.

History

1820

— 4 September: news of Ørsted's compass reaches the Paris Academy; Ampère presents his first analysis within two weeks and demonstrates that parallel currents attract.

1826

— Ampère's Mémoire delivers the complete mathematical theory of currents — the circulation law in embryo.

1855

— Maxwell's first field paper recasts Ampère's results in the language of lines of force.

1861

— Maxwell adds the displacement current, extending the circuital law to changing fields and predicting electromagnetic waves.

1968

— Tokamak results stun plasma physics: toroidal magnetic bottles, designed by circulation-law bookkeeping, become fusion's leading architecture.

Going further

Texts. NCERT Physics Class 12, Ch. 4 (§4.7) · HC Verma Vol. 2, Ch. 35 · SL Arora, ISC Physics XII, magnetic effects of current.

Provenance. All results standard. A verifier should recompute: circulation $\mu_0 \times 1.0\ \mathrm{A} = 1.26\times10^{-6}$ T·m and $\mu_0 \times 5\ \mathrm{A} = 6.3\times10^{-6}$ T·m; wire field $2\times10^{-7}\times6.0/0.040 = 3.0\times10^{-5}$ T; cable profile $B_{\max} = 1.0$ mT at $R = 2.0$ mm with 10 A, halves $0.50$ mT at $r = R/2$ and $r = 2R$ ; branch continuity at $r = R$ .

Prerequisites. Fields of Straight Wires & Circular Loops · Gauss's Law · The Biot–Savart Law. Leads to. Solenoids & Toroids (next in this chapter) · Ampère Setups — Cables, Sheets & Cavities (coming in the E4 problem gym) · Ampère's Law (UG deep dive).

Is Ampère's law more fundamental than the Biot–Savart law, or just more convenient?

For steady currents they are exactly equivalent — each can be derived from the other, and every field the slice-recipe builds obeys the circulation law automatically. So at this level: two costumes, one physics; choose by convenience (symmetry → Ampère, everything else → Biot–Savart). At university, the ranking shifts: the circulation law, upgraded by Maxwell's displacement-current term, becomes one of the four Maxwell equations — the actual laws of electromagnetism — while Biot–Savart remains a solution recipe valid only for steady currents. The habit of thinking in circulations is therefore an investment: it is the form of the law that survives.

Why do outside currents cancel so perfectly around the loop? It feels too good.

It is the $1/r$ falloff doing precision bookkeeping. An outside wire's field is strong on the near arc of your loop and weak on the far arc — but the near arc, seen from the wire, is short, and the far arc is long. For a field falling exactly as $1/r$ , "stronger × shorter" and "weaker × longer" balance exactly on any closed path that doesn't enclose the wire. The circulation law and the wire's $1/r$ law are two faces of one geometric fact — just as outside charges contribute zero net flux through a closed surface, the same conspiracy in inverse-square clothing.

What exactly counts as "enclosed" current? The loop is a curve, not a box.

Stretch an imaginary soap film across the loop, any way you like — flat, domed, billowing. A current is enclosed if it punctures the film, with sign from the right-hand rule, counted net over all its punctures. For steady currents the choice of film provably cannot change the answer (current that enters a closed region must leave it), which is why nobody ever draws the film in practice — but the picture rescues you in tricky geometries: a hairpin that dips through and back counts zero; a wire coiled so it pierces the film five times counts five times, which is precisely why an N-turn coil in next module's solenoid argument contributes N times its current.

The law is "always true" — so why can't it find the field of a finite wire segment?

Because true and solvable are different things. For a finite segment (say, with feed wires bending away), the circulation around a centred circle still equals $\mu_0 I_{\text{enc}}$ — the law holds. But the field on that circle is not the same at every point (points near the segment's middle feel more than points off its ends), so the sum refuses to collapse to $B \times 2\pi r$ ; one equation faces many unknowns and extracts none of them. The information isn't gone — it's just not liftable by this tool. Biot–Savart slice-summing, indifferent to symmetry, handles it. First question on any new problem: can I say the symmetry sentence honestly? If not, change tools.

Where does the solenoid fit into all this?

It is the law's greatest hit, and it opens the next module. Wrap wire into a long tube of $n$ turns per metre and choose a rectangular loop straddling the wall: symmetry silences three of the four legs, the film is pierced $nL$ times, and out falls $B = \mu_0 n I$ — the uniform interior field behind electromagnets, relays and MRI bores. The same bookkeeping shows a toroid's field never leaves the doughnut: every turn pierces an outside film once in, once out — net zero, the hairpin rule at scale. Three lines each, in Solenoids & Toroids.

Does the law still work if the current changes with time?

Not as written — and the repair is one of the great stories ahead. For steady currents, "current through any film on the loop" is unambiguous. Let the current change — say, while charging a capacitor — and different films can honestly disagree: one film crosses the wire and counts $I$ ; another slips between the capacitor plates, where no charge flows, and counts zero. Maxwell's 1861 fix adds a second source term — the displacement current, made from changing electric fields — that restores agreement and, as a by-product, predicts electromagnetic waves. That story is the finale of this track (Displacement Current, in the EM Waves chapter); for every circuit in this chapter, currents are steady and the law as stated is exact.

How is this law related to Gauss's law — are they the same trick?

Same species, different organ. Both are totals laws: a global sum (flux through a closed surface; circulation around a closed loop) constrained by a count of sources (charge; current), and both become two-line solvers exactly when symmetry reduces the field to one number on the probe. But they measure different anatomy: Gauss tests how fields stream outward — electric fields do, from charges. Ampère tests how fields wrap around — magnetic fields do, around currents. Run the probes crosswise and you get electromagnetism's two quiet zeros: magnetic flux out of any closed surface is zero (no magnetic charges — Gauss's Law for Magnetism, in the next chapter), and static electric fields have zero circulation (the fact behind potential energy).