The Ampere-Maxwell law

Ampere's law ties the circulation of B around a closed loop to the current through a surface bounded by that loop — and it never says which surface, because for steady currents the choice cannot matter. Charge accumulates nowhere, so whatever current enters a closed region leaves it, and every surface spanning the loop is pierced the same number of times. Now build a circuit whose current stops: a parallel-plate capacitor with circular plates of radius 5.0 cm, charging at I = 2.0 A. Take a circular loop of radius 3.0 cm around the lead. Span it with a flat disc and the lead pierces it, giving ∮B⋅dl = μ₀ × 2.0 A = 2.51 × 10⁻⁶ T m. Span the same loop with a bag-shaped surface that bulges past the plate and closes inside the gap, and no charge crosses it at all, giving zero. Both surfaces are legal. The law as Ampere wrote it contradicts itself the moment charge piles up somewhere.

Follow the charge. Current arriving at the plate has nowhere to go, so dq/dt = I. Between the plates that charge sets up a field E = σ/ε₀ = q/(ε₀A), uniform if the gap is small, so the electric flux through the part of the bag lying inside the gap is Φ_E = EA = q/ε₀. Differentiate: ε₀ dΦ_E/dt = dq/dt = I. The bag was never empty after all — the conduction current it misses is replaced, ampere for ampere, by a growing electric flux. Adding μ₀ε₀ dΦ_E/dt to the right-hand side therefore makes the two surfaces agree: the disc collects μ₀I from the lead and nothing from flux, the bag collects nothing from current and μ₀I from flux. The same argument works for any loop and any circuit, because it is only charge conservation written twice. For this capacitor, with A = 7.85 × 10⁻³ m², the field in the gap is climbing at dE/dt = I/(ε₀A) = 2.9 × 10¹³ V m⁻¹ s⁻¹.

The repaired law earns its keep as a computing tool exactly where Ampere's law did: when symmetry forces B to be constant in magnitude and tangential along the loop. Between circular plates the charging field lies along the axis and is uniform, so the arrangement is symmetric about that axis and B must circle it with constant magnitude. Choose a circle of radius r centred on the axis, midway between the plates. Then ∮B⋅dl = 2πrB. Inside the rim the loop's surface catches only the flux passing through area πr², so the enclosed displacement current is I(r/R)². With I = 2.0 A and R = 5.0 cm, a loop at r = 3.0 cm encloses 0.72 A, and 2πrB = μ₀(0.72 A) gives B = 4.8 μT. Push r past the rim and the whole 2.0 A is enclosed, so B = μ₀I/(2πr), the field of the wire. The two expressions meet at r = R, where B reaches its largest value, 8.0 μT.

Signs here are a convention you must apply consistently, not a physical extra. Pick a direction to walk the loop when evaluating ∮B⋅dl. Your right hand, fingers curling that way, points the thumb along the normal n̂ of the spanning surface. That single n̂ then grades both source terms: a conduction current counts as positive when it flows along n̂, and the flux is Φ_E = ∫E⋅n̂ dA, so dΦ_E/dt is positive when E grows along n̂. Reverse the walking direction and all three quantities change sign together, leaving the physics untouched. The payoff is a real prediction. While the capacitor charges, E in the gap grows from the positive plate towards the negative one, and B circles that direction by the right-hand rule, in the same sense as around the lead. Let the capacitor discharge and E shrinks, dΦ_E/dt turns negative, and B in the gap reverses — even though no charge has ever crossed the gap.

The two source terms are rarely comparable. For a sinusoidal field of amplitude E₀ and angular frequency ω in a material of conductivity σ and permittivity ε, the conduction current density is σE₀ and the displacement current density is εωE₀, so their ratio is ωε/σ. Inside copper, σ = 5.96 × 10⁷ S m⁻¹ and ε ≈ ε₀, so at 60 Hz that ratio is 5.6 × 10⁻¹⁷: the displacement term in household wiring sits seventeen orders of magnitude below the conduction term, which is why Ampere's law survived unamended for decades. The ordering reverses in poor conductors and at high frequency, the two terms matching at f = σ/(2πε). In seawater, σ ≈ 4 S m⁻¹ and ε ≈ 81ε₀, putting that crossover near 8.9 × 10⁸ Hz. In a vacuum gap σ is zero, so the displacement term is not merely dominant, it is the only term there is.

Set the two circulation laws side by side. Faraday gives ∮E⋅dl = −dΦ_B/dt; the Ampere-Maxwell law gives ∮B⋅dl = μ₀I_enc + μ₀ε₀ dΦ_E/dt. The flux terms mirror each other, and that reciprocity is what the rest of this unit is built on. Two asymmetries survive on purpose. There is a conduction term here and none in Faraday, because isolated electric charge exists while no magnetic charge has ever been found, so there is no magnetic current to put in. And Faraday carries a minus sign where this law does not, which is Lenz's law: an induced effect opposes the change that made it, while a growing electric flux drives B in the positive sense. The constant on the new term repays a second look too. μ₀ε₀ = 1.113 × 10⁻¹⁷ s² m⁻², which is 1/c² — a speed assembled from two constants measured with capacitors and coils.