Using Ampere's circuital law,derive the expression for the magnetic field at a perpendicular distance $r$ from an infinitely long current-carrying wire. Explain it.

(N/A) Consider an infinitely long straight wire carrying a current $I$. To find the magnetic field at a distance $r$ from the wire,we choose a circular Amperean loop of radius $r$ centered on the wire.
The magnetic field lines due to a straight current-carrying wire are concentric circles. Therefore,at any point on the loop,the magnetic field $\vec{B}$ is tangential to the loop.
From the symmetry of the setup,the magnitude of the magnetic field $B$ is constant at all points on the loop. Thus,the line integral of the magnetic field is:
$\oint \vec{B} \cdot d\vec{l} = \oint B dl \cos 0^{\circ}$
$= B \oint dl$
$= B(2\pi r)$
According to Ampere's circuital law:
$\oint \vec{B} \cdot d\vec{l} = \mu_{0} I$
Equating the two expressions:
$B(2\pi r) = \mu_{0} I$
Therefore,the magnetic field is:
$B = \frac{\mu_{0} I}{2\pi r}$