State ampere’s circuital law and express it mathematically.

Ampere’s law states that the path integral or line integral over a closed loop of the magnetic field produced by a current distribution is given by

here I refers to the current enclosed by the loop.

Ampere’s law is a useful relation that is analogous to Gauss’s law of electrostatics. It is a relation between the tangential component of magnetic field at points on a closed curve and the net current through the area bounded by the curve.

To evaluate the expression for

, let us consider a long, straight conductor carrying a current I, passing through the centre of a circle of radius r in a plane perpendicular to the conductor.

According to Biot-Savart law of magnetism, the field has a magnitude

at every point on the circle, and it is tangent to the circle at each point.

The line integral of

around the circle is

Since

is the circumference of the circle,

Therefore,