A current does not need to be touched to be felt. Run charge down a straight wire and a magnetic field wraps it in perfect circles — and the strength of that whole embrace is fixed by one number: the current threading the loop. Walk any closed path around the wire, add up B along it, and you always get the same total, μ₀I, no matter how wide you walk. Down the center, data flows: the current goes in, the engine wraps the field, the invariant loop integral comes out. The blue team builds it; the red team breaks it.
source Ampère, Théorie des phénomènes électro-dynamiques (1826); circulation form in J. C. Maxwell, A Treatise on Electricity and Magnetism (1873) — archive.org/details/electricandmagne01maxwrich. No single stable "law" page in the original text — AMBER. Rendered, not quoted.
Two statements, one truth. Local: around an infinite straight wire the field is azimuthal with magnitude B = μ₀I / (2πr) — it falls as 1/r, so doubling r halves B (not quarters it — that is Coulomb's field, not this one).
Global: the line integral of B around any closed loop equals μ₀ times the current threading it — ∮B·dl = μ₀Ienc. On a circle of radius r that is (2πr)·B = μ₀I, and the r cancels: the total is radius-independent.
For the current form, the loop integral at four radii — same total every time:
| r (cm) | B (µT) | ∮B·dl (µT·m) |
|---|
Ampère's loop integral is the-gauss-law's twin: Gauss sums flux through a closed surface to find enclosed charge; Ampère sums circulation around a closed loop to find enclosed current. One is the divergence of the field, the other the curl.
And a moving version of this circle is exactly what feeds the-faraday-law: change the current, change the flux, and an EMF is born. Each sphere is the next one's premise.
The blue team's live check: re-run the engine over a sweep of loop radii and confirm the loop integral is invariant and equals μ₀I. If red tampers the falloff, this badge is where it shows.
Three inputs decide everything: the current I threading the wire, the radius r of the loop you walk, and the constant of the vacuum, μ₀ = 4π×10⁻⁷ T·m/A. The wire is infinite and straight; the loop is a circle centred on it.
By symmetry B has the same magnitude everywhere on that circle and points along it — so the messy line integral collapses to a single product, (2πr)·B. That collapse is the whole trick, and it is what you feed the panel below.
Slide the loop out and watch B fall as 1/r — while the loop integral (2πr·B) refuses to move.
Circles = the field lines; arrowheads = azimuthal direction (right-hand rule, current out of the page). Arrow length ∝ B, thinning as 1/r. The white ring is your chosen loop.
Change any control — B and the loop integral are computed live from μ₀I/(2πr), never looked up.
What the machine produces, proven: whatever the loop radius, ∮B·dl = μ₀Ienc. The field B changes with r; the loop integral does not. A loop that encloses no current returns exactly zero. And B scales linearly with I — double the current, double the field.
The blue team's witness (left) confirms the invariance live; the red team (right) tries to make the total wobble.
The clean closed form B = μ₀I/(2πr) also holds only under high symmetry — an infinite straight wire. For a finite wire, a bent wire, or a coil edge you must integrate Biot–Savart; Ampère's law is still true but no longer solves for B by itself. "Symmetry" is a load-bearing assumption, not decoration.
"B falls off as 1/r², like the force between charges." Cut. That is Coulomb/Gauss. The field around a wire falls as 1/r — inverse-linear. The engine proves it: B(2r)/B(r) = ½, not ¼.
"A bigger loop measures a bigger circulation." Cut. The loop integral is μ₀Ienc — independent of the loop's size and shape. Widen the loop and B drops by exactly the amount the path lengthens.
"Ampère's law is one of Maxwell's, untouched." Kept, corrected. Ampère gave the magnetostatic case; Maxwell added the displacement-current term. The fourth equation is Ampère–Maxwell, not Ampère alone.
The red team's move: rewrite the falloff as 1/r² (an inverse-square field, as if the wire were a point charge). Now the loop integral 2πr·B = μ₀I/r depends on the radius — the invariant is broken. The blue team's witness (window 7) is watching.
Force inverse-square and the loop integral stops being radius-independent — the witness sweeps the radii, sees the total drift, and turns red. Nothing is faked; the attack is real and it is caught.