Self Induction:

 Consider a coil 'L' wound over a hollow wooden cylinder [in Fig. 1]. Soft iron is packed into a hollow region to increase magnetic flux linked with 'L' (since 'μ' for iron is greater than one). The two free terminals of the coil are connected to a source of e.m.f. 'E' through a tap key 'K'. 

Fig. 1. Self induced e.m.f. in a coil. 

     As key 'K' is pressed, current flowing through the circuit starts increasing. As it increases (at make) from zero to maximum, an induced e.m.f. is set up in it due to the phenomenon of electromagnetic induction. 

     According to Lenz's law, the induced e.m.f. opposes the change (increasing current) which produces it. Therefore, its direction is opposite to that of current. When current achieves its maximum value, magnetic flux linked with the coil becomes maximum (constant). No induced e.m.f. is there in the coil at this stage. When key is released (at break) current decreases from maximum to zero. This results in decrease of magnetic flux linked with the coil. Again an e.m.f. is set up. The direction of e.m.f., according to Lenz's law, should be same as that of the current. Thus, it is clear, that induced e.m.f. induced e.m.f., whenever it appears in the circuit, opposes a change in the strength of current flowing through the circuit. This property of the circuit is called self induction. 

     Self induction of a circuit is defined as the property of the circuit, by virtue of which it tends to oppose a change in the strength of current, through it, by inducing an e.m.f. in itself. 

Experimental demonstration of self induction

     Property of self induction can be demonstrated by connecting a bulb 'B' across the two terminals of the coil. The e.m.f. induced in the coil comes directly across the bulb and causes a strong current to flow through it. Therefore, the bulb gives a flash of light, whenever induced e.m.f. is present. A bright flash in the bulb at make and at break of the circuit testifies the property of self induction. 

Co- efficient of Self Induction

Let 'ΦB' be the magnetic flux linked with a circuit due to a current 'I' flowing through it. 

         ΦB ∝ I           or      ΦB = LI       .... (1) 

Where 'L' us called the 'Co-efficient of self induction' of the circuit. It depends upon :

(i) area of cross-section of coil

(ii) number of turns of the coil

(iii) nature of material packed within the coil. 

If          I = 1, ΦB = L

     Co-efficient of self induction of a circuit is defined as the magnetic flux linked with it when a unit current flows through it. 

     Differentiating equation (1) with respect to 't', we get

      ΦB /dt = d/dt (LI) = L dI/dt    [∵  L does not depends upon time]

     According to Faraday's law of electromagnetic induction, 

               E = -  ΦB/dt

Where E = e.m.f. induced in the circuit due to a current changing at the rate dI/dt, 

          - E = L dI/dt       

or        E = - L dI/dt                   ......(2) 

If         dI/dt = 1, L = [E]

     Therefore, co-efficient of self induction is also defined as the e.m.f. induced in the circuit, due to a current changing at a unit rate in itself. 

Unit of 'L'

     (i) In S.I. co-efficient of self induction of a circuit is measured in 'henry'. 

     In equation (2) 

     If     dI/dt = 1 A s-1, E = 1 V, then L = 1 H. 

     Co-efficient os self induction of a circuit is said to be 1 henry if an e.m.f. of 1 V is induced in it due to a current changing at the rate of 1 amp s-1 in itself. 

       1 henry= 1 V/1 A s-1 = 1 Vs A-1  

     (ii) In C.G.S. (e.m.u.) system. In the C.G.S. system (e.m.u.) co-efficient of self induction is measured in 'e.m.u. of inductance' or 'abhenry'. 

            |L| = E/dI/dt

∴   1 e.m.u. of inductance = 

       1 e.m.u. of potential difference/1 e.m.u. of current/1 second

     Co-efficient of self induction of a circuit is said to be 1 e.m.u. of inductance if an e.m.f. of 1 e.m.u. is induced in it due to a current changing at the rate 1 e.m.u. s-1.

Dimensions of L

From equation (2) 

         [L] = E/dI/dt

               = [M L2 T-3 A-1]/[A1 T-1]

∴      [L] = [M1 L2 T-2 A-2]

So, the dimension of L are 1, 2, -2, -2 in mass, length, time and electric current respectively. 

Relation between 'henry' and e.m.u. of inductance

      1 henry = 1 volt/1 amp/sec

                     = 10⁸ e.m.u. of potential difference/1/10 e.m.u. of current/sec

∴    1 henry = 10⁹ e.m.u. of inductance. 

Self Induction of Solenoid

A coil wound over an insulating cylinder and having some self induction can be treated as  a solenoid carrying current. Magnetic field at any point, on the axis of a solenoid is

        B = μ0μr  N/l I

     Therefore, magnetic flux linked with one turn of coil 

                  = B × A = μ0μr N/l I × A

Where 'A' is the cross-sectional area of the coil. Total magnetic flux ΦB linked with whole of coil (of N-turns), 

                 ΦB = μ0μr N/l  I × A × N

or             ΦB = μ0μr N² I A/l

     Therefore, induced e.m.f. E is 

             E = - dΦB/dt = - μ0μr N² A/l  dI/dt    ... (3) 

     Comparing equations (2) and (3), 

       L = μ0μr N² A/l = μ0μr n NA

Where 'n' is the number of turns per unit length. 

     Thus, co-efficient of self induction of a coil depends upon following factors :

1. Area of the coil. Greater the area if coil, greater is the co-efficient of self induction. 
2. Number of turns. Greater the number of turns, greater is co-efficient of self induction. 
3. Relative permeability of the core. A coil having soft iron as its core has a greater co-efficient of self induction than that having air core.

IMPORTANT NOTES Resistance 'R' is a measure of opposition to the electric current (constant) while self inductance 'L' is a measure of opposition to a change in current in a circuit.  Self induction is often termed as 'electrical inertia' since it is analogous to inertia in mechanics. Inertia in mechanics opposes a change in velocity while self induction (electrical current).  An ideal inductor is considered to be a coil of zero resistance while an ideal resistor (straight wire) is considered to be possessing zero self induction. Both, ideal inductor and ideal resistor cannot be obtained practically.  An ideal inductor is equivalent to a closed key in an electrical circuit.  Wires used in resistance boxes have a special type of winding called non-inductive winding. The wire is first of all doubled in itself (Fig. 2) and then it is wound over a mica sheet. When current flows through it, the direction of current in two windings situated close to each other will be opposite to each other. Thus, the magnetic field produced by them will get neutralised. Since there is no resultant magnetic field in the region, there will no induced e.m.f. in the wire when current in the wire is changing. Thus, the only opposition will be of resistive nature.  Fig. 2.

Inductances in series and parallel

Like resistors, inductors can also be connected in series and parallel. While connecting them we shall suppose that the two are so spaced apart that changing current in one does not affect the second. 

 

(i) Inductances in series

     Consider two inductances L₁ and L₂ connected in series as shown in Fiɡ. 3. Let I be the instantaneous current flowinɡ throuɡh each of them . Let E₁ and E₂ be the values of e.m.fs. induced in the first and second coil respectively. 

Fig. 3. Inductances in series. 

          E₁ = - L₁ dI/dt   and    E₂ = - L₂ dI/dt

   

     If 'E' is the net e.m.f. induced in the combination

                E = E₁ + E₂

     ∴         E = (- L₁ dI/dt) + (- L₂ dI/dt) 

or            E = - (L₁ + L₂) dI/dt                      ... 4

     If L = net inductance of the combination, 

                E = - L dI/dt                                 ... 5

     Comparing equations (4) and (5), we get

                L = L₁ + L₂

     Thus, the resultant inductance of two inductors in series with each other is equal to the sum of their individual inductances. 

(ii) Inductances in parallel

     When the two inductances are connected in parallel with each other, they carry different electric currents but have same potential difference across their ends. 

Fig. 4. Inductances in parallel. 

     If I₁ + I₂ are the instantaneous currents flowing through them (Fig. 4), the e.m.f.'s induced in them are given by

               E₁ = - L₁ dI₁/dt

and        E₂ = - L₂ dI₂/dt

     Since    E₁ = E₂ = E          (Resultant e.m.f.) 

     ∴             E = L₁ dI₁/dt = - L₂ dI₂/dt        ... 6

     Since the inductances are connected in parallel, 

                     I = I₁ + I₂

     ∴       dI/dt = dI₁/dt + dI₂/dt                  ... 7

     From equation (6), 

             dI₁/dt = - E/L₁    and    dI₂/dt = - E/L₂

     Substituting in equation (7), 

            dI/dt = (- E/L₁) + (- E/L₂) 

or        dI/dt = - E (1/L₁) + (1/L₂)              ... 8

     If L is the resultant inductance of the combination

                  E = - L dI/dt

or        dI/dt = - E × 1/L                            ... 9

     Comparing equations (8) and (9), 

   

                 1/L = 1/L₁ + 1/L₂.

     Thus, when two inductances are connected in parallel with each other, the reciprocal of the resultant inductance is equal to the sum of the reciprocal of individual inductances. 

     It may be noted that the inductances combine in manner similar to the resistance.