Explain Ampere's circuital law.

(N/A) Ampere's circuital law provides an alternative and appealing way to express the relationship between a magnetic field and the current that produces it.
Consider an open surface with a boundary. The surface has a current passing through it.
We consider the boundary to be made up of a number of small line elements. Consider one such element of length $d\vec{l}$.
We take the value of the tangential component of the magnetic field $B_{T}$ at this element and multiply it by the length of that element $dl$:
$B_{T} dl = \vec{B} \cdot d\vec{l}$
As the number of elements increases,the sum tends to a line integral.
Ampere's circuital law states: The line integral of the magnetic field $\vec{B}$ around any closed loop is equal to $\mu_{0}$ times the total current $I$ passing through the surface enclosed by the loop.
Mathematically,$\oint \vec{B} \cdot d\vec{l} = \mu_{0} \Sigma I$
Where $\Sigma I$ is the algebraic sum of the currents enclosed by the loop.
The sign convention for the current is determined by the right-hand rule: If the fingers of the right hand are curled in the direction of the loop integration,then the thumb points in the direction of the positive current.