Hello Om,

Ampere's circuital law states that the integral lines of the magnetic field B around any closed circuit is equal to (permeability constant) times the total current 'I' passing through this closed circuit.

Mathematically;

B.dl= *I

Proof for a straight current carrying conductor:

Consider a long straight current carrying conductor 'I'. According to Biot-Savart law, the magnitude of the magnetic field B due to the current carrying conductor at any point at a distant 'r' from it is mathematically given by;

B= *I*2*r

The magnetic field B is directed along the circumference of the circle of radius 'r' with the wire as center. The magnitude of the field B is same all points on the circle. To evaluate the line integral of the magnetic field B along the circle, we consider a small current element dI along the circle. At every point on the circle, both B and dl are tangential to the circle so that the angle between them is zero.

B.dI= B*dl cos(0)= B*dl (1) = B*dl

Hence the line integral of the magnetic field along the circular path is

B.dl= B*dl= B (dl)= *I*2*r*I = *I*2r*2r

B.dl=*I

This proves Ampere's law. This law is valid for any assembly of current and for any arbitrary closed loop.

Hope it helped. Thank you.
Best of luck.