Self Inductance
Consider a loop of a conducting material carrying electric current. The current produces a magnetic field B. The magnetic field gives rise to magnetic flux. The total magnetic flux linking the loop is
dφ = B.ds
In the absence of any external source of magnetic flux (for example, an adjacent coil carrying a current), the Biot-Savart’s law tells that the magnetic field and hence flux will be proportional to the current (I) in the loop,
φ ∝ I or φ = LI
where L is called self-inductance of the coil. The circuit elements which oppose change in current are called inductors. These are in the form of coils of varied shapes and sizes. The inductance of an indicator depends on its geometry.
Faraday’s Law in terms of Self-Inductance
If current in a loop changes, the magnetic flux linked through it also changes and gives rise to self–induced emf between the ends. In accordance with Lenz’s law, the self-induced emf opposes the change that produces it.
To express the combined form of Faraday’s and Lenz’s Laws of induction in terms of L,
ε = – dφ/dt
ε = – L dI/dt
= – L (I2 - I1)/t
where I1 and I2 respectively denote the initial and final values of current at t = 0 and t = τ.
Unit of self-inductance is ohm-second, An ohm-second is called a henry, (abbreviated H).
The self-induced emf is also called the back emf. The back emf in an inductor depends on the rate of change of current in it and opposes the change in current. Moreover, since an infinite emf is not possible, an instantaneous change in the inductor current cannot occur. Thus, current through an inductor cannot change instantaneously.
Self-inductance of Solenoid
Consider a long solenoid of cross-sectional area A and length l, which consists of N turns of wire. Using Ampere’s law to determine magnetic field of a long solenoid:
|B| = µ0nI
where n = N/L denotes is the number of turns per unit length and I is the current through the solenoid.
The total flux through N turns of the solenoid is
φ = N|B|A = µ0N2AI/L
Self-inductance of the solenoid is
L = φ/I = µ0N2A/L