Magnetic field due to current carrying solenoid :
Consider a very long solenoid having n turns per unit length of solenoid.Let current I be flowing through the solenoid.
The magnetic field inside the solenoid is almost uniform , strong and directed along.
The axis of the solenoid. The magnetic field outside a very long solenoid is very weak and can be neglected.
Step 1 : Let P be a point well within the solenoid. Consider any rectangular loop ABCD ( known as Amperian Loop ) passing through P.
Then
$\oint \overrightarrow{B}.\overrightarrow{dl}$ = Line integral of magnetic field across the loop ABCD,
$\oint \overrightarrow{B}.\overrightarrow{dl}= \int_{A}^{B} \overrightarrow{B}.\overrightarrow{dl} + \int_{B}^{C} \overrightarrow{B}.\overrightarrow{dl} + \int_{C}^{D} \overrightarrow{B}.\overrightarrow{dl} + \int_{D}^{A} \overrightarrow{B}.\overrightarrow{dl}$
$\overrightarrow{B}$ is perpendicular to paths BC and AD i.e. angle between $\overrightarrow{B}$ and $\overrightarrow{dl}$BN is 90° for these paths.
$\int_{D}^{A} \overrightarrow{B}.\overrightarrow{dl} = \int_{B}^{C}\overrightarrow{B}.\overrightarrow{dl} = \int B.dl.cos 90^{0} = 0$
Since path CD is outside the solenoid, where $\overrightarrow{B}$ is taken as zero , so
$\int_{C}^{D} \overrightarrow{B}.\overrightarrow{dl}= 0$
For path AB , the direction of $\overrightarrow{B}$ and $\overrightarrow{dl}$ is same $\theta = 0^{0}$
$\oint \overrightarrow{B}.\overrightarrow{dl} = \int_{A}^{B} B.dl.cos 0^{0}$
$\oint \overrightarrow{B}.\overrightarrow{dl} = B\int_{A}^{B}dl$
$\oint \overrightarrow{B}.\overrightarrow{dl} = Bl$...(i)
Step 2 : According to Ampere's circuital law
$\oint \overrightarrow{B}.\overrightarrow{dl} = \mu_{0} $× net current enclosed by loop ABCD
$\oint \overrightarrow{B}.\overrightarrow{dl} = \mu_{0} $× number of turns in the loop ABCD × I
$\oint \overrightarrow{B}.\overrightarrow{dl} =\mu_{0}nlI$...(ii)
Step 3 : comparing equation (i) and (ii) , we get
$Bl = \mu_{0}nlI$
$B= \mu_{0}nI$
Since $n = \frac{N}{l}$
Where N = total number of turns of solenoid
$B = \frac{\mu_{0}NI}{l}$