Similar to Gauss’ law for electric fields, Ampere’s law is for magnetic field, which states that $\oint_c \vec B \cdot \vec{ds} = \mu 0 I{enc}$

• Line integral of $\vec B$ over the closed path $C$., also called the circulation of $\vec B$ ****is equal to $\mu _0$ multiplied by the current enclosed by the closed path (loop)
• The closed path (loop) is called the Ameriean loop (AL)
  • $\vec B$ is the magnetic field
  • $\vec {ds}$ is the differential line element along the AL
    • Tangential to the path at each point along the path
  • $\mu _0$ is free space permeability (currents reside in vacuum)
  • $I_{enc}$ is the current enclosed by the AL
  • Note that $\vec B \cdot \vec {ds}$ is the projection of $\vec B$ along the path (AL) at each point on the path
    • $\vec B \cdot \vec {ds} = B ds \cos \theta$

Applications of Ampere’s Law (on paper)

Solenoids (on paper)

Capacitors were used to store electric energy and produce a uniform $\vec E$ in the case of the parallel plates. Solenoids are used to store magnetic energy and produce a uniform magnetic field.

• Conductor wrapped in a shape
• The clsoed path $a-b-c-d$ is the Ampereian loop of choice, for this loop we have $B = \mu _0 nI$, the magenetic field inside a solenoid