Ampère's circuital law

Remarks

The two forms are equivalent by Stokes' theorem. The left side is the line integral of $B$ around $\partial S$. The right side is $\mu_0$ times the flux of $J$ through $S$, the current through $S$. The proof writes $B = \nabla \times A$ with $A = \frac{\mu_0}{4\pi} \int J(y) / \abs{x - y} \, dy$, and uses $\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \Delta A$. Here $\Delta A = -\mu_0 J$ by Poisson's equation, and $\nabla \cdot A = 0$ because $\nabla \cdot J = 0$. Without $\nabla \cdot J = 0$ the law fails, since $\nabla \cdot (\nabla \times B) = 0$ would force $\nabla \cdot J = 0$. For time-dependent fields Maxwell added the displacement current: $\nabla \times B = \mu_0 J + \mu_0 \varepsilon_0 \partial_t E$. Example: for a long straight wire with current $I$, a circle of radius $s$ around the wire gives $2 \pi s \abs{B} = \mu_0 I$, so $\abs{B} = \mu_0 I / (2 \pi s)$.

Going deeper

For a student

Ampère's law is the magnetostatic analogue of Gauss's law: with symmetry it gives fields without integration. It yields the field $\mu_0 n I$ inside a long solenoid with $n$ turns per unit length, and the field of a toroidal coil. With Maxwell's correction it becomes one of Maxwell's equations, and the correction is exactly what makes the equations consistent with charge conservation.