Explain Ampere's Law and Biot-Savart's Law, including their mathematical formulations and use cases.

3.1: Ampere's Law and Biot-Savart's Law

Ampere's Law states that the line integral of the magnetic field $\vec{B}$ around any closed loop is proportional to the total current $I_{enc}$ passing through the loop. It is useful for calculating magnetic fields in highly symmetric situations.

$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$$

where $\mu_0$ is the permeability of free space.

Biot-Savart's Law describes the magnetic field $d\vec{B}$ produced by an infinitesimal current element $I d\vec{l}$. It