Long Current-Carrying Solenoid

The solenoid magnetic field is the vector sum of the field produced by the individual turns that make up the solenoid

Magnetic field B is nearly uniform and parallel to the axis of the solenoid at interior points near its center and external field near the center is very small

Consider a dashed closed path abcd as shown in figure .Let l be the length of side ab of the loop which is parallel to the is of the solenoid

Let us also consider that sides bc and da of the loop are very-very long so that side cd is very much far away from the solenoid and magnetic field at this side is negligibly small and for simplicity we consider its equal to 0

At side a magnetic field B is approximately parallel and constant. So for this side $∫B.dl=Bl$

Magnetic field B is perpendicular to sides bc and da ,hence these portions of the loop does not make any contributions to the line integral as B.dl=0 for the side bc and da

Side cd lies at external points solenoid where B.dl=0 as B=0 or negligibly small outside the solenoid

Hence sum around the entire closed path reduces to Bl

If N are number of turns per unit length in a solenoid then number of turns in length l is nl.The total current through the rectangle abcd is NIl and from ampere 's law

$Bl=μ_0NlI$ or $B=μ_0NI$