Explain how Biot-Savart's law enables one to express the Ampere's circuital law in the integral form , viz.,
`oint vecB .vec(dl) = mu_0 I`
where I is the total current passing through the surface.

Explanation

Let us consider an infinitely long straight wire in which a current I is flowing. To find magnetic field at a point P situated at a normal distance R from the wire consider a circular loop of radius R around the wire and in a plane perpendicular to the wire so that magnetic field `vecB` is tangential to the circumerence of the circle. For such a case
`oint vecB. vec(dl) = int B dl = B int dl = B . 2 pi R`
But as per Biot-Savart.s law we know that magnetic field B at a normal distance R from a long long straight current carrying wire is
`B = (mu_0 I)/(2 pi R)`
Hence, we get :
`oint vecB . vec(dl) = ((mu_0 I)/(2 pi R)) 2 pi R = mu_0 I`.
which is the integral form of Ampere.s circuital law.