(a) State Ampere's circuital law, expressing it in the integral form.
(b) Two long coaxial inlsulated soenoids, `S_(1)` and `S_(2)` of equal lengths are would one over the other as shown in the figure. A steady current ''I'' flows through the inner solenoid `S_(1)` to the other end B, which is connected to the outer solenoid `S_(2)` through which the same current ''I'' flows in the opposite direction so as the come out at end A. If `n_(1)` and `n_(2)` are the number of turns per unit length, find the magnitude and direction of the net magnetic field at a point
(i) inside on the axis and
(ii) outside the combined system.

Explanation

(a) Ampere's circuital law states that the circulation of the resultant magnetic field along a closed plane curve is equal to time the total current crossing the area bounded by the closed curve, provided the electric field inside the loop remains constant.
In the above illustration, the Ampere's circuital law can be written as follow :
(b) (i) The magnetic field due to a current carrying solenoid :
where, N = number of turns per unit length i = current through the solenoid Now, the magnetic field due to solenoid `S_(1)` will be in the upward direction and the magnetic field due to `S_(2)` will be in the downward direction (by right-hand screw rule).
In the upward direction.

(ii) The magnetic field is zero outside a solenoid.