6.9 Inductance
An electric current can be induced in a coil by flux change produced by another coil in its vicinity or flux change produced by the same coil. These two situations are described separately in the next two sub-sections. However, in both the cases, the flux through a coil is proportional to the current. That is,
Further, if the geometry of the coil does not vary with time then,
For a closely wound coil of N turns, the same magnetic flux is linked with all the turns. When the flux
he constant of proportionality, in this relation, is called inductance. We shall see that inductance depends only on the geometry of the coil and intrinsic material properties. This aspect is akin to capacitance which for a parallel plate capacitor depends on the plate area and plate separation (geometry) and the dielectric constant K of the intervening medium (intrinsic material property).
Inductance is a scalar quantity. It has the dimensions of
6.9.1 Mutual inductance
Two long co-axial solenoids of same length l.
Consider Fig. 6.15 which shows two long co-axial solenoids each of length l. We denote the radius of the inner solenoid
When a current
For these simple co-axial solenoids it is possible to calculate
where
Note that we neglected the edge effects and considered the magnetic field
We now consider the reverse case. A current
The flux due to the current
where
We have demonstrated this equality for long co-axial solenoids. However, the relation is far more general. Note that if the inner solenoid was much shorter than (and placed well inside) the outer solenoid, then we could still have calculated the flux linkage
We explained the above example with air as the medium within the solenoids. Instead, if a medium of relative permeability µ r had been present, the mutual inductance would be
It is also important to know that the mutual inductance of a pair of coils, solenoids, etc., depends on their separation as well as their relative orientation.
Now, let us recollect Experiment 6.3 in Section 6.2. In that experiment, emf is induced in coil
It shows that varying current in a coil can induce emf in a neighbouring coil. The magnitude of the induced emf depends upon the rate of change of current and mutual inductance of the two coils.
6.9.2 Self-inductance
In the previous sub-section, we considered the flux in one solenoid due to the current in the other. It is also possible that emf is induced in a single isolated coil due to change of flux through the coil by means of varying the current through the same coil. This phenomenon is called self-induction. In this case, flux linkage through a coil of N turns is proportional to the current through the coil and is expressed as
where constant of proportionality L is called self-inductance of the coil. It is also called the coefficient of self-induction of the coil. When the current is varied, the flux linked with the coil also changes and an emf is induced in the coil. Using Eq. (6.15), the induced emf is given by
Thus, the self-induced emf always opposes any change (increase or decrease) of current in the coil.
It is possible to calculate the self-inductance for circuits with simple geometries. Let us calculate the self-inductance of a long solenoid of cross- sectional area A and length l, having n turns per unit length. The magnetic field due to a current I flowing in the solenoid is
where nl is the total number of turns. Thus, the self-inductance is,
If we fill the inside of the solenoid with a material of relative permeability
The self-inductance of the coil depends on its geometry and on the permeability of the medium
The self-induced emf is also called the back emf as it opposes any change in the current in a circuit. Physically, the self-inductance plays the role of inertia. It is the electromagnetic analogue of mass in mechanics. So, work needs to be done against the back emf (
If we ignore the resistive losses and consider only inductive effect, then using Eq. (6.16),
Total amount of work done in establishing the current I is
Thus, the energy required to build up the current I is,
This expression reminds us of
Consider the general case of currents flowing simultaneously in two nearby coils. The flux linked with one coil will be the sum of two fluxes which exist independently. Equation (6.9) would be modified into
where
Therefore, using Faraday’s law,