State Ampere's Circuital Law. Use it to derive the magnetic field along the axis of a current carrying solenoid of length 'l' having 'n' number of turns.

Definition:

The circulation of the resultant magnetic field along a closed, plane curve is equal to μ₀ times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant. Mathematically it means 

Magnetic field due to solenoid: 

Consider a rectangular amperian loop PQRS near the middle of solenoid as shown in figure where PQ = l.

Let the magnetic field along the path PQ be B and zero along RS. As the paths QR and PS are perpendicular to the axis of solenoid, the magnetic field component along these paths is zero. Therefore, the path QR and PS will not contribute to the line integral of magnetic field B. 

Total number of turns in length l = Nl 

The line integral of magnetic field induction B over the closed path PQRS is

here,

and

also, ouside the solenoid (as B = 0)

thus, we have

now, from Ampere's Circuital Law

= μ₀ × Number of turns in rectangle × Current BL = μ₀nLI 

so, finally we get

This relation gives the magnetic field induction at a point well inside the solenoid.