Ampere's Law and Its application in Solenoid Part 1

Ampère's Law Statement: The line integral of the magnetic field (\(B\)) around any closed path (Amperian loop) is proportional to the total electric current (\(I\)) passing through the area enclosed by the path.  Mathematical form: \(\oint \mathbf{B}\cdot d\mathbf{s}=\mu _{0}I\)  \(\oint \mathbf{B}\cdot d\mathbf{s}\) is the line integral of the magnetic field around the closed loop.  \(\mu _{0}\) is the permeability of free space.  \(I\) is the net current enclosed by the loop.  Applications Long current-carrying wire: Used to find the magnetic field at a distance from a long straight wire. The formula is \(B=\frac{\mu _{0}I}{2\pi r}\).  Solenoid: Determines the uniform and strong magnetic field inside a long solenoid, where the field inside is approximated as \(B=\mu _{0}nI\) (n = number of turns per unit length).  Toroid: Calculates the magnetic field inside a toroid.  Long cylindrical conductor: Used to find the magnetic field inside and outside a current-carrying conducting cylinder.  Electromagnets, motors, and generators: Provides a foundational principle for the design of these and other electrical devices