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Magnetic Field Inside a Long Solenoid

Consider a long straight solenoid having n turn per unit length. Let I be the current flowing in the solenoid. It is experimentally noted that magnetic field outside the solenoid is very small in comparison with the field inside. The lines of induction inside the solenoid are straight and parallel.
Consider a closed path pqrs. The line integral of magnetic field B along path pqrs is

Φpqrs B. dl = ∫pq B. dl + ∫qr B. dl + ∫qr B. dl + ∫rs B. dl + ∫sp B. dl        … (1)

Let pq = l. For path pq, B and dl are along same direction.
. :    ∫pq B. dl = ∫ Bdl = Bl

For paths qr and sp, B and dl are mutually perpendicular.
. :     ∫qr B. dl = ∫ B.dl = ∫ B dl cos 90 = 0

For paths rs, B = 0 (since field is zero outside a solenoid)
. :    ∫rs B . dl = 0

Eq. (1) becomes, Φpqrs B. dl = ∫pq B. dl = Bl                … (2)

By Ampere’s law, Φ B. dl = μ0 x net current enclosed by path
            Bl = μ0 (nl)i
. :            B = μ0  ni                    … (3)

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