Ampère's law

In vacuum with prescribed stationary currents, the magnetostatic field obeys ∇×B=μ₀J and ∇·B=0. Choose an oriented surface S whose boundary is the closed contour C. Orient the tangent on C by the right-hand rule about the chosen normal n̂ of S. Stokes' theorem relates the curl integral over S to the circulation around C.

An enclosed current is a SIGNED flux of J through this oriented surface; reversing the orientation of both C and S changes both signs and preserves the equation.

The left-hand side is a signed integral of B's TANGENTIAL component. Neither the integral's value nor its vanishing determines B at each point without additional information. In particular, a uniform nonzero externally imposed B has zero integral around a closed curve and no enclosed current.

The expression B(2πs)=μ₀I is valid only when source symmetry proves that B is everywhere tangent to a circle of radius s and has constant magnitude on that circle. Selecting a circular integration path does not manufacture symmetry in an off-center or irregular current distribution.

Take a +z-directed infinite current filament of strength I through the origin, surrounded by vacuum and without other sources. Translation along z, rotation about z and reflection symmetry imply B=Bφ(s) φ̂, independent of azimuth and z. Choose C as a coaxial circle of radius s>0, traversed counterclockwise as viewed from +z; then its associated normal points +z and I_enclosed=+I.

On C, B·dℓ=Bφ(s)s dφ. Integrating φ from 0 to 2π yields the field obtained independently by the Biot–Savart integral in Chapter 02. For a finite straight wire the requisite translation symmetry fails near its endpoints, so this simplification cannot be imposed there.

Replace the zero-radius filament by a long circular conductor of radius a carrying uniform stationary current density J=I/(πa²) ẑ. The same cylindrical symmetry now holds for ideal long uniform current, but I_enclosed changes with the radius s of the Ampèrian circle.

For 0≤s<a, only the area πs² contributes: I_enclosed=Jπs²=I(s²/a²). For s>a the enclosed current is I. The two field expressions agree at s=a in the absence of a singular surface-current sheet, and B→0 as s→0. Using the exterior 1/s formula at the axis would falsely introduce the ideal-filament singularity.

Let an isolated infinitely extended plane z=0 carry uniform surface current K=K x̂ in vacuum, with no other sources. Planar translation symmetry and reflection-related source geometry give equal-magnitude opposite y-directed magnetic fields above and below. A rectangular Ampèrian loop in the yz plane, with its long sides of length L on opposite sides of the sheet and a normal +x̂, encloses I_enclosed=KL.

With a right-hand-oriented traversal, the two long sides each contribute BL; the short sides contribute zero because their tangents are along z. Consequently 2BL=μ₀KL. The direction is B(z>0)=−μ₀K/2 ŷ and B(z<0)=+μ₀K/2 ŷ. Ampère's law alone gives their DIFFERENCE; the opposite-field symmetry is an additional independent premise.

Place an Ampèrian circle in a uniform external magnetic field B₀. If it encloses no current, its total circulation is zero because the positive and negative tangent projections cancel. B₀ remains finite everywhere. Likewise, two currents of opposite sign can have zero algebraic sum through a surface while producing nonzero local fields on most of its boundary.

The field of a current distribution must be obtained from a genuine symmetry simplification, Biot–Savart, a boundary-value solution, or an equivalent method. Do not infer B=0 merely from I_enclosed=0.

For time-varying charge density, charge continuity reads ∂ρ/∂t+∇·J=0. Taking the divergence of the purely stationary relation ∇×B=μ₀J would incorrectly require ∇·J=0 at every instant; the left side is identically divergence-free. Maxwell's vacuum correction includes electric displacement current.

The charging-capacitor example makes the issue concrete: a spanning surface cutting the wire captures conduction current, while a bulging surface passing between plates can capture no conduction current but has changing electric flux. The Ampère–Maxwell law gives consistent circulation for both surfaces. This does not license applying the static Biot–Savart or static cylindrical-conductor results to arbitrary time-varying circuits.

A long cylinder has a=2.00 mm, I=3.00 A uniform over its cross-section. At s=1.00 mm (<a), B=μ₀Is/(2πa²)≈1.50×10⁻⁴ T=150 μT. At s=a, B≈300 μT. At s=4.00 mm (>a), B≈150 μT; the interior field rises linearly, whereas the exterior field falls inversely with s.

For an isolated infinite sheet K=10.0 A/m, each side has magnitude μ₀K/2≈6.28 μT. Dimensional checks: [μ₀I/s]=T and [μ₀K]=T. Neither number describes the near-edge field of a finite sheet or a short current-carrying cylinder.

Identify a stationary current distribution, the chosen contour direction and spanning-surface orientation. Use the geometry of ALL sources, not the appearance of a hand-picked integration path, to establish the direction and constancy of B·t̂.

Determine the enclosed current with its correct sign and measure (filament I, volume J·dA, or sheet K times intercepted length), and write the full circulation equation before extracting an unknown field. Check current reversal, units, continuity at material boundaries and agreement with Chapter 02 wherever both laws apply.

1. For a +z infinite filament I=2.00 A, evaluate B at radius 0.0200 m, give the signed direction at x>0, and check Biot–Savart.
2. For a=2.00 mm and I=3.00 A uniformly across a long cylinder, derive I_enclosed at s=1.00 mm and compute B at s=1.00, 2.00 and 4.00 mm.
3. An infinite isolated sheet at z=0 carries K=10.0 A/m along +x. Compute B above and below, including signs.
4. A closed circular contour lies in a nonzero uniform external B₀ and encloses no current. What is its circulation, and may one conclude B₀=0?
5. Two equal parallel +z infinite wires at x=−d/2 and +d/2 carry equal I. Determine B at their midpoint. What changes if one current is reversed?
6. During capacitor charging, two surfaces span the same contour, one intersecting a wire and the other passing between the plates. Which term restores a consistent circulation?

Chapter 04 constructs the magnetic vector potential A with B=∇×A, addresses gauge freedom, and recovers the Biot–Savart kernel for stationary sources.