Ampere's circuital law can be stated as :
The line integral of the magnetic field around some closed loop is equal to the times the algebraic sum of the currents which pass through the loop.
Proof:
Consider a long straight conductor carrying current i in upward direction as shown below in the figure.
Now from Biot Savart law, the magnetic field at any point P which is at a distance R from the conductor is given by
\[dB=\frac{\mu _{0}i}{2\pi R}\]
The line integral of B around the circle is
\[\oint Bdl=\oint \frac{\mu _{0}i}{2\pi R}dl=\mu _{0}i \ (Since \oint dl=2\pi R)\]
\[\therefore \oint Bdl=\mu _{0}i\]
This is the same result as stated by Ampere law
