Ampère's Law

Ampère's law in integral form is universally true for steady currents, but it only computes B analytically when symmetry makes B constant in magnitude (and either parallel or perpendicular to dℓ) along a chosen Amperian loop. Three high-symmetry cases dominate textbooks: cylindrical symmetry (infinite straight wire, coaxial cable, solid cylindrical conductor), planar symmetry (infinite current sheet), and translational symmetry along an axis (infinite solenoid, toroid). Without that symmetry — finite solenoid edges, square loops, helical coils — you must fall back on Biot-Savart integration. Ampère's law still holds; you just can't extract B from the integral by inspection.

Original Ampère's law works for closed steady currents but fails at gaps. Consider a charging capacitor with current I in the leads: an Amperian loop around the wire has I_enc = I if the bounding surface intersects the wire, but I_enc = 0 if the surface bulges out and passes between the capacitor plates. Same loop, two different answers. Maxwell (1861) resolved the contradiction by adding ε₀ dΦ_E/dt — the displacement current — to I_enc. Between the plates, even though no charge crosses, the changing electric flux contributes ε₀ dΦ_E/dt that exactly equals the conduction current outside. With this term, ∮ B·dℓ = μ₀(I_enc + ε₀ dΦ_E/dt), and the law becomes consistent for any time-varying field — which immediately predicts that E and B can sustain each other as a propagating wave.

Inside an infinite ideal solenoid, B is uniform along the axis and zero outside. Choose an Amperian loop that's a rectangle of length L, with one side along the axis inside, one side outside, and two short legs perpendicular to the axis. The integral splits: along the inside leg, ∮ B·dℓ = BL; along the outside leg, B = 0 so the integral vanishes; the perpendicular short legs contribute zero because B ⊥ dℓ. Total: ∮ B·dℓ = BL. The enclosed current is the number of turns crossing the rectangle: nL turns each carrying I, so I_enc = nLI. Equating: BL = μ₀nLI, giving B = μ₀nI. The length L cancels — only the turns-per-meter density n matters.

Inside an infinite ideal solenoid, B = μ₀nI is uniform, axial, and surprisingly large for modest currents. Outside, B = 0. This near-perfect confinement is unique among basic geometries — neither a single loop nor a straight wire produces field that drops to zero outside a region. For a finite solenoid the picture deteriorates: near the ends the field weakens by a factor of 1/2 on the central axis at the mouth, fringing fields leak outward, and external B is small but nonzero. Long solenoids (length much greater than radius) approximate the ideal case well across most of their interior. This combination of strong, uniform interior field plus vanishing exterior field makes solenoids the workhorse of electromagnets, MRI scanners, and laboratory field standards.

Wrap a solenoid into a torus and you get B confined entirely inside the donut, falling off as 1/r where r is distance from the central axis. For a toroid with N total turns each carrying I, choose a circular Amperian loop of radius r inside the windings, concentric with the central axis. By symmetry, B is tangential and constant along the loop: ∮ B·dℓ = B(2πr). The enclosed current is NI (every turn pierces the loop once). Equating: B = μ₀NI/(2πr). Outside the donut and inside the central hole, B = 0. Toroidal geometry is preferred for: tokamak fusion reactors (no end-leakage to confine plasma), ferrite EMI chokes, current transformers, and high-Q inductors.

Ampère's law in integral form is ∮ B·dℓ = μ₀ I_enc, where the enclosed current is the total current piercing any surface bounded by the loop. For consistency, the answer must not depend on which surface you choose. Take a charging capacitor with current I in the lead wire. Pick an Amperian loop encircling the wire near the capacitor. A flat disk surface intersects the wire: I_enc = I, so the law gives ∮ B·dℓ = μ₀I. Now pick a hemispherical surface that bulges through the gap between the plates: it intersects no current at all, so I_enc = 0 and the law gives ∮ B·dℓ = 0. Same loop, same B. Contradiction. Maxwell's displacement current ε₀ dΦ_E/dt patches this: in the gap, the time-rising E flux produces a virtual current density that exactly equals the conduction current outside, restoring surface independence.