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Ampere's law states $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$: the circulation of $\vec{B}$ around any closed path equals $\mu_0$ times the net current threading through the loop. The sign convention uses the right-hand rule — curl fingers along the direction of path traversal, and the thumb indicates positive current direction.

While universally valid, Ampere's law is practically useful only when symmetry allows $B$ to be pulled out of the integral. The three classic applications: (1) Long straight wire — circular Amperian loop gives $B = \mu_0 I/(2\pi r)$. (2) Solenoid — rectangular loop gives $B = \mu_0 nI$ inside, zero outside. (3) Toroid — circular loop gives $B = \mu_0 NI/(2\pi r)$ inside, zero outside.

Key subtlety: $\oint \vec{B} \cdot d\vec{l} = 0$ does NOT mean $B = 0$ everywhere on the loop — only that positive and negative contributions cancel. Conversely, $B \neq 0$ on a loop doesn't require enclosed current; external currents create fields but their line integral contribution vanishes. This distinction between the field at a point and the line integral is a frequent source of JEE conceptual errors.