Inductance of a coil | Concept of Self Induction | Solenoid

When current in a coil changes, there is a change in magnetic flux linking the coil. Hence, an emf is induced in the coil. This is called self induced emf (e_s)  and the process is called self-induction. According  to Lenz’s law, the direction of this induced emf is  such that it opposes the cause that has produced it. Now the cause of e_s  is the change in  magnetic flux linking the coil or current through the coil. Hence induced emf will oppose the change of current in the coil.

Self-Inductance of a coil

The property of a coil by virtue of which it opposes the change of current in the coil is called Self-Inductance.

Consider a coil of N turns carrying a current I. Let the flux linked with each turn of the coil be Φ. Then the flux linkages of the coil is NΦ. 

If the current changes, the flux linkages of the coil will also change. This will set up a self induced emf (e_s) in the coil given by

e_s = -(d(NΦ)/dt)

Since flux is due to the current in the coil, it follows that flux linkages will b directly proportional to the current I   i.e.   NΦ ∝ I

Therefore          e_s ∝ -(dI/dt)

Or                     e_s = -L (dI/dt)     ,  Where L is a constant of proportionality and is called Self-Inductance or simply Inductance or coefficient of Self-Induction. Unit of  Inductance is Henry (symbol,H).

Definition of Henry :- The inductance of a coil is said to be 1 Henry if current changing at the rate of 1 ampere per second through the coil induces an emf of 1 Volt in it. Or

Since      NΦ=LI 

Therefore, inductance of a coil is said to be 1 Henry if a current of 1 ampere in the coil sets up a total flux of 1 weber in it.

Derivation for Inductance of a coil / long solenoid :- 
Consider a long solenoid of length l , number of turns N and area of cross-section A  is carring a current I.

The magnetic field inside the solenoid is given by  B=(u₀NI)/ l

The magnetic flux linking any turn of the coil is    Φ=BA cos 0⁰ = (u₀NIA)/ l

Therefore, the magnetic flux linkage is    NΦ = (u₀N² IA) / l

Also, NΦ=LI  , giving us

L=(u₀N² A) / l

Clearly we can see inductance like capacitance depends  upon geometrical factors such as number of turns , cross-section area and length of the coil.

The above expression is for air as the medium in the coil. In case there is a material inside the coil having relative permeability u_r , then self inductance of coil becomes

L=(u₀ u_rN² A) / l