Ampere's Law and Choosing a Loop

Ampere's law reads: the closed loop integral of B dot dl equals mu0 times I_enc, the current piercing the surface bounded by the loop. Walk any closed path through a magnetic field and add up how much B points along your direction of travel at each step. That running total depends only on how much current is stitched through the loop, not on the shape of the loop or how far it sits from the wires. It is the magnetic cousin of Gauss's law: Gauss's law uses a closed surface to catch flux from enclosed charge, Ampere's law uses a closed loop to catch circulation from enclosed current. Neither field has to be uniform in general, but both laws become genuinely useful only when symmetry lets you pull B out of the integral.

Picking the loop is the whole skill. You want a path where B has constant magnitude and stays parallel (or perpendicular, contributing zero) to dl along the entire loop, so the integral collapses to B times the loop length. For a long straight wire, the field circles the wire in loops of constant magnitude, so a circle centered on the wire, lying in the plane perpendicular to it, is the natural choice: B is the same everywhere on that circle and always tangent to it. Then the integral is just B(2 π r), and Ampere's law gives B(2 π r) = mu0 I, so B = mu0 I / (2 π r), done in three lines.

Current outside the loop still creates a nonzero B on the loop, but it contributes zero net circulation: its field lines loop around that outside wire and pass through your loop's boundary in a way that cancels over the full trip. Only current actually piercing the bounded surface counts toward I_enc, which is why choosing the loop also means choosing what you count.