Self Inductance

Consider the figure given below
When we establish a current through an inductor or coil, it generates a magnetic field and this result in a magnetic flux passing through the coil as shown in figure 1(a).
If we vary the amount of current flowing in the coil with time, the magnetic flux associated with the coil also changes and an emf ξ is induced in the coil.
According to the Lenz's law, the direction of induced emf is such that it opposes its cause i.e. it opposes the change in current or magnetic flux.
This phenomenon of production of opposing induced emf in inductor or coil itself due to time varying current in the coil is known as self induction.
If I is the amount of current flowing in the coil at any instant then emf induced in the coil is directly proportional to the change in current i.e.

where L is a constant known as coefficient of self induction.
If (-dI/dt)=1 then ξ=L
Hence the coefficient of self induction of a inductor or coil is numerically equal to the emf induced in the coil when rate of change of current in the coil is unity.
Now from the faraday's and Lenz's laws induced emf is

comparing equation 1 and 2 we have,

or Φ=LI
Again for I=1, Φ=L
hence the coefficient of self induction of coil is also numerically equal to the magnetic flux linked with the inductor carrying a current of one ampere
If the coil has N number of turns then total flux through the coil is
Φ_tot=NΦ
where Φ is the flux through single turn of the coil .So we have,
Φ_tot=LI
or L=NΦ/I
for a coil of N turns
In the figure given below consider the inductor to be the part of a circuit and current flowing in the inductor from left to right
Now when a inductor is used in a circuit, we can use Kirchhoff�s loop rule and this emf(Self induced emf) can be treated as if it is a potential drop with point A at higher potential and B at lower potential when current flows from a to b as shown in the figure
We thus have
V_ab=LdI/dt

Self induction of a long solenoid

Consider a long solenoid of length l, area of cross-section A and having N closely wound turns.
If I is the amount of current flowing through the solenoid them magnetic field B inside the solenoid is given by,
Magnetic flux through each turn of the solenoid is,