What is mutual induction
Suppose that the two coils, P and S are placed closed to each other so that they are electrically insulated from each other but are magnetically coupled. The coils P and S are respectively called primary coil and secondary coil. A battery of emf E and a tapping key K are connected to the primary coil P while a galvanometer G is connected across the secondary coil as shown in Fig- 1. Now we press the tapping key K to allow the electric current to flow in the primary coil. The current starts growing exponentially and after some finite time t, it will attain a maximum value I0 as shown in Fig -2.
As the current grows in the primary coil, the increasing magnetic field will be set up in it as per the laws of magnetic effects of current and so the increasing magnetic flux will link the primary coil itself. Due to the phenomenon of self-induction, an emf is induced in the primary P coil itself in such a direction that it opposes the growth of battery current as per Lenz’s law
Since the secondary coil S is placed closed to the primary coil P, the changing magnetic flux also starts building up with the S- coil as it is magnetically coupled with P- coil however both the coils are electrically insulated with each other. Therefore, an induced emf is developed in the S- coil also in such a way that it opposes the growth of current in the primary coil.
Since the secondary coil S is placed closed to the primary coil P, the changing magnetic flux also starts building up with the S- coil as it is magnetically coupled with P- coil however both the coils are electrically insulated with each other. Therefore, an induced emf is developed in the S- coil also in such a way that it opposes the growth of current in the primary coil.
Direction of Induced emf/Induced current in mutual Induction
In both the cases when the battery current grows or decays in the primary coil, the changing magnetic flux links the S- coil and emf is induced in it. As a result, the induced current flows in the secondary coil.
The induced current opposite to the battery current when key is pressed.
Coefficient of Mutual induction
Let us suppose that at any instant of time, an electric current I is passing through the primary coil P. The magnetic flux linking the secondary coil is found to be directly proportional to the current passing through the primary coil at that moment of time, thus
Here, M is constant of proportionality and is called as coefficient of mutual induction or simply mutual inductance of the two coils, if I = 1unit current, then from eqn. 01, we have,
Thus, the coefficient of mutual induction of two coils is defined as the magnetic flux linked with one coil when an electric current of one-unit flows through the other coil placed in its neighbors. From Faraday’s second Law of electromagnetic induction, the induced emf (e) developed in secondary coil due to mutual induction will be
Therefore, the coefficient of mutual induction/ mutual inductance of two coils is defined as the induced emf in one coil, when the electric current changes in the nearby coil at the rate of one unit current per second.
S.I. Units of Mutual Inductance
Using equation 03, we write, M = e/(dI/dt), If e = 1 volt, dI/dt =1 ampere/ second, then M = 1 volt/.(1ampere second-1 ) = 1 Henry ( H )
Thus, the coefficient of mutual induction or inductance of two coils is said to be one henry, when one volt of an emf in induced in one coil due to an electric current changes in the neighboring coil at the rate of one ampere per second.
factors affecting Mutual inductance between the two coils
There are various factors that affect the mutual inductance of two coils. The shape and size of the two coils, magnetic permeability of the material of the core on which the coils are wound, and the separation between them greatly affect the mutual inductance between the coils. The coefficient of coupling also affect the mutual induction between the two coils significantly which in turn depends upon the orientation of the coils relative to each other, Thus, K =( M/L1 L2 )1/2 , where K is called coefficient of Coupling between the coils and L1 & L2 be their self inductances
When two coils are wound are wound on each other
The coefficient of coupling is maximum when the two coils P (primary) and S (secondary) are wound on each other and so the mutual induction between the coils will be maximum. The mutual inductance further increases if the two coils are wound on the soft iron core because the magnetic permeability of soft iron is higher and entire magnetic flux originated from the primary coil links to the secondary coil. This is the reason that the transformer coils are wound on soft iron core
When the two coils are placed along their common axis
The coefficient of coupling between the two coils will still be significantly large when the two coils are placed along their common axis but less than that in case wound on each other as in previous case. Therefore, the mutual inductance of the two coils will be less when placed along their common axis than that when wound on each other and also on soft iron core.
When two coils are placed with their axes perpendicular to each other
The coefficient of coupling will be minimum in the case when the two induction coils are placed such that their axes are mutually perpendicular to each other as shown in the figure. Therefore, the entire magnetic flux originated at the primary coil would not couple with the secondary coil. Thus, the mutual induction of the two coils will be minimum in this case. This type of coupling is not recommended.
Mutual Inductance of two long solenoid
Let us suppose two long solenoids S1 and S2 each of equal length L, such that S2 completely fits S1 with in itself and N1 and N2 be the number of turns per unit their length as shown in fig- 7. Let us suppose that an electric current I is made to pass through the solenoid S1 so that a magnetic field will be set up with in itself as per the principles of magnetic effect of currents will be given by
As a result, the magnetic flux originated at S1 will link with the solenoid S2 . This magnetic flux linking the solenoid S2 is directly proportional to the electric current I1 passing through the solenoid S1 . Thus,
Here, M21 represents the mutual inductance or coefficient of mutual induction of the two solenoids S1 and S2 . It is evident that when an electric current passes through a solenoid, the magnetic lines of force remain confined to the space with in the solenoid only. Therefore, the magnetic field in the annular region between the the cross-sections of the two solenoids is zero when the current is passed through the solenoid S1 .
Therefore, the magnetic flux linked with the each turn of the solenoid S2 will be equal to B1 times the area of cross-section S1. Now the magnetic flux linked with the each turn of the solenoid S2 will be equal to B1 A, where A is the area of cross-section of solenoid S1. Now the total magnetic flux linked with the solenoid S2 will be
Exactly following the same steps, we can evaluate the mutual Inductance of the two solenoids, under the consideration that an electric current I2 passes through the solenoid S2. Thus we shall find M12 that come to be
We analyze that the coefficient of mutual induction/ the inductance of the two long solenoids remains same irrespective of the fact that the electric current is passed through the inner solenoid or the outer solenoid, Thus M21 = M12. Thus, in general the mutual inductance of the two long solenoid is represented as
Coefficient of mutual induction of two concentric coils
Suppose that C₁ and C₂ are the two circular coils having same common center O (Concentric Coils), of radii r₁ and r₂ such that r₂ > r₁. Let us further suppose that an electric current I₂ is passed through the coil C₂, then the magnetic field produced by this coil at its center will be
This equation 12 gives the required result about the coefficient of mutual induction of two concentric coils of different radii.
Summary of mutual induction
- When the two coils are coupled inductively, the induced emf is developed in each coils due to the phenomenon of self induction in addition to the emf induced due to the phenomenon of mutual induction.
- The coefficient of mutual induction/mutual inductance of the two coils each having self inductances L1 and L2 is given by the relation M=K2 L1 L2, where, K is constant for the couple of the given coils and is called coefficient of coupling.
- The coefficient of mutual inductance between the two coils wound over each other does not depend upon the area of cross-section of the outer coil irrespective of the fact that the current is passed through which coil, inner coil or outer coil.