Let consider a solenoid of N turns with length l and area of cross section A. It carries a current I. If B is the magnetic field at any point inside the solenoid, then

Magnetic flux per turn = B × area of each turn

But, **B = μ _{0}NI / l**

Magnetic flux per turn = **μ _{0}NIA / l**

Hence, the total magnetic flux (φ) linked with the solenoid is given by the product of flux through each turn and the total number of turns.

**φ = (μ _{0}NIA / l) × N**

i.e **φ = (μ _{0}N^{2}IA / l) … … (1)**

If L is the coefficient of self-induction of the solenoid, then

φ = LI … … (2)

From equations (1) and (2)

**LI = (μ _{0}N^{2}IA / l)**

**∴**** L = (μ _{0}N^{2}A / l)**

If the core is filled with a magnetic material of permeability μ, then, L **= (μN ^{2}A / l)**