Self Inductance:

You know that while current flows in a coil, flux is generated. If the current is changing with time, the flux linked with the coil is also time varying and an electromotive force (e.m.f.) is induced given by the law

e ∝ di/ dt

or,

e = L (di /dt)

In Eq. (6), the constant of proportionality L is termed as self-inductance of the coil and its unit is Henry.

According to Faraday's law of electromagnetic induction, the induced e.m.f. in a coil with N turns is specified by

e = N (d φ /dt)

 From Eqs. (6) and (7), we may write

L (di / dt )= N( d φ/ dt)

or, L = N (d φ/ dt)

As Φ versus i is linear, Eq. (8) may be expressed as

L = N (φ /i)

 Since we know from Eq. (2.5) that

φ= Ni / S = Ni / (l/ μ₀ μ_r A) =( N μ₀ μ_r A / l )i

 So,     φ / i = N μ₀ μ_r A/ l

 From Eqs. (9) and (10) we now obtain

L = N μ₀ μ_r A/ l = N²  μ A/ l

Where    μ = μ₀ μ_r