AMPÈRE'S LAW

The Biot–Savart integral is honest. Hand it any current distribution and it grinds out the magnetic field at any point. It is also exhausting — for symmetric geometries you spend a page deriving a one-line answer.

The history rhymes here. Coulomb's law gives a true answer for any charge distribution; Gauss's law gives the easy answer when the geometry has symmetry. Magnetism needed the same trick. Ampère found it in 1826, a few years after Ørsted's 1820 discovery that current makes a compass deflect. The shape is the same as Gauss's idea for E: a closed integral on the left, the source on the right, the geometry collapsed away.

The cost is the same too. The law is always true; it is only useful — solves for B in two lines — when the geometry lets you pull B out of the integral. Three geometries do: the long straight wire, the long solenoid, and the toroid. Those three are most of magnetostatics in practice.

Pick a closed loop in space — any shape, anywhere. Walk around it once and, at every point, take the component of B that lies along your direction of travel. Add up those components, weighted by the small step you took. The total is the closed line integral of B around the loop.

That phrase has its own symbol: the integral sign with a small circle through it, ∮, signals "the integration path is a closed curve, returning to where it started." When you read ∮ B · dℓ aloud, say "the line integral of B around the loop." Each tiny step dℓ is a vector pointing along the path; the dot product picks out only the part of B that runs along the path, ignoring any part that crosses it. Send the loop in the opposite direction and every dℓ flips sign — you get the same magnitude with the opposite sign.

The other piece of the equation is the current enclosed. Imagine the loop as the wire rim of a soap film. The "current enclosed" is the total electric current threading through any surface that fills in the loop. Currents that punch through the film count; currents that loop back outside count zero. Two wires going opposite directions cancel.

The constant μ₀ is the vacuum permeability — magnetism's analogue of ε₀, also a single number measured in the lab. The shape of the loop is missing from the right-hand side: stretch it, dent it, fold it into a star — as long as you do not cross any current, the integral is unchanged. Outside currents contribute exactly zero, because every line entering the loop also leaves.

The sign comes from the right-hand rule. Curl the fingers of your right hand in the direction you walk the loop; your thumb points in the direction "positive" current must flow to count as positive enclosed.

Take an infinite straight wire carrying current I and ask for the field at perpendicular distance r. By the cylindrical symmetry, B at radius r has the same magnitude everywhere on a circle of radius r around the wire, and points tangent to that circle (right-hand rule: thumb along I, fingers curl in the direction of B).

Choose that circle as your Amperian loop. Then B is parallel to dℓ everywhere on it, and the integrand B · dℓ is just B · dℓ, with B constant. The integral collapses to B · 2π·r. Setting that equal to μ₀·I:

Two lines instead of the half-page Biot–Savart needs. The scene above does the rest of the proof: switch the Amperian loop from a circle to a square to a wobbly blob, and the numerical line integral of B · dℓ tracks μ₀·I exactly. Slide the loop off the wire and the integral drops to zero, because you no longer enclose the source.

A long, tightly-wound coil of wire — a solenoid — produces a uniform magnetic field along its axis on the inside, and a near-zero field outside. To prove it, draw a rectangular Amperian loop straddling the wall: one long side of length L lies inside the solenoid, parallel to the axis; the other long side lies far outside, also parallel; the two short sides cross the wall, perpendicular to the field.

Inside, B is uniform and points along the long side, so that side contributes B·L to ∮B·dℓ. Outside, B ≈ 0, so the parallel side outside contributes zero. The two short sides are perpendicular to B inside, and B is zero outside, so they contribute zero. The loop encloses every turn that lies between its two long sides. With n turns per metre, that is n·L turns, each carrying current I. So:

L vanishes — the field magnitude is set by the density of turns, not their total count. Plug in n = 1000 turns/m and I = 1 A and you get 1.26 mT, a tenth of the Earth's field. Real solenoids — MRI scanners, particle-accelerator focusing magnets, doorbell relays — are this geometry, possibly with iron in the core to multiply the field a few thousand times.

Bend the solenoid into a doughnut and you have a toroid. The toroidal symmetry says B at a point inside the doughnut runs in a circle concentric with the central axis, with the same magnitude all the way around that circle.

Choose that circle as your Amperian loop, at radius r from the axis. The line integral collapses to B · 2π·r. Every one of the N total turns punches through the disk bounded by the loop exactly once, so the enclosed current is N·I:

The field is not uniform inside the toroid: it falls off as 1/r, so the inner edge of the doughnut sees a stronger field than the outer edge. Outside the toroid, an Amperian circle encloses no net current — every turn punches in once and out once — so B = 0 there. This is why toroids are the geometry of choice for confining hot plasmas: a tokamak is a giant toroid whose internal B-field traps charged particles on circular orbits.

The integral form ∮B·dℓ = μ₀·I_enc is a global statement: walk around a loop, sum up the field. There is also a local version, an equation that holds at every point in space.

The local version uses an operator called the curl. For a vector field, curl asks: at this point, how much does the field circulate around the point itself? Imagine inserting a tiny paddlewheel; the curl tells you how fast it spins, and about which axis. A field that flows in straight lines has zero curl; a field that winds around a point has non-zero curl, pointing along the wheel's axis. We write the curl of B as ∇×B; see curl for the full unpacking.

The local form of Ampère's law says the curl of B at a point equals μ₀ times the local current density J at that point:

This is incomplete. Lorentz-era electromagnetism noticed it predicts no field around a charging capacitor between the plates, where no actual current flows but the electric field is changing. Maxwell will fix this in §07 by adding a "displacement current" term ε₀ ∂E/∂t, which restores consistency and — almost as a free bonus — produces light.

The three closed-form solutions of Ampère's law are what made electromagnetic engineering possible. The straight-wire formula sets the safe spacing of high-current bus bars. The solenoid formula sizes electromagnets, the focusing coils of every cathode-ray tube, the main coils of an MRI machine, and the actuators of every electric door lock. The toroid formula confines the plasma in a tokamak fusion reactor — the donuts at JET and ITER are toroidal coils whose interior B-field traps deuterium ions on closed paths long enough to fuse.

The pattern keeps recurring. A symmetric closed integral. One source on the right. The geometry collapses, the algebra is one line. Coulomb has Gauss; Biot–Savart has Ampère.