Ampere's law

Printed on the AP Physics C: Electricity and Magnetism sheet, topic 12.4 in the CED, where the closed path is called an Amperian loop. It is the magnetic counterpart of Gauss's law, and it fails in the same way: the statement is always true, and useless for finding $B$ unless you can argue the field has constant magnitude and a fixed angle along the path you chose.

Symmetry is the whole technique. Choose a loop on which $\vec{B}$ is either constant and parallel to $d\vec{\ell}$ or perpendicular to it, so the integral collapses to $B \times$ (length) or to zero. For a long straight wire the loop is a circle, giving $B = \mu_0 I / (2\pi r)$; for a long solenoid it is a rectangle with three inert sides, giving $B_{\text{sol}} = \mu_0 n I$. Both are labelled derived equations, so you are expected to produce them.

The CED's boundary statement limits quantitative work to symmetric fields, naming long straight wires, long solenoids, and conductive slabs or cylindrical conductors carrying a current density. Where the symmetry is absent, the Biot-Savart law is the tool instead.

$I_{\text{enc}}$ counts only current threading the loop, with sign. Current outside contributes nothing to the integral, even though it does contribute to $\vec{B}$ at points on the loop.

As Maxwell's fourth equation it gains a changing-electric-field term, which the CED does not expect you to use. Nothing here is on the AP Physics 2 sheet.