Ampère's Law

Ampere's law states that the circulation of the total magnetic field around any closed path equals $\mu_0$ times the current threading it: $\oint\vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$. It is always true, and it hands you $B$ only when symmetry makes the field constant in magnitude and everywhere tangent along the chosen loop, as it does for an infinite wire, an ideal solenoid, and a toroid. The enclosed current is a signed census: currents piercing along the right-hand thumb count positive, each turn of a coil counts separately, and inside a uniform thick conductor only the fraction $r^2/R^2$ threads the loop. A current loop in a uniform field feels zero net force and a torque $\vec{\tau} = \vec{\mu}\times\vec{B}$ with $\vec{\mu} = NIA\hat{n}$.

Four errors dominate. Erasing an outside current's field from the integrand, when what vanishes is its net circulation and not its field. Factoring $B$ out of the integral around a loop where the field varies, which turns a true statement into a false formula. Miscounting the enclosed current by adding magnitudes without signs, dropping the turn count of a solenoid, or using the full current inside a thick wire instead of the enclosed fraction. And giving a loop in a uniform field a net translational force, or putting maximum torque at the orientation where the torque is actually zero.