# Mutual Inductance

**Mutual Inductance** is the ratio between induced Electro Motive Force across a coil to the rate of change of current of another adjacent coil in such a way that two coils are in possibility of flux linkage.

Mutual induction is a phenomenon when a coil gets induced in EMF across it due to rate of change current in adjacent coil in such a way that the flux of one coil current gets linkage of another coil.
Mutual inductance is denoted as ( M ), it is called co-efficient of Mutual Induction between two coils.

**Mutual inductance** for two coils gives the same value when they are in mutual induction with each other.
Induction in one coil due to its own rate of change of current is called self inductance (L), but due to rate of change of current of adjacent coil it gives **mutual inductance** (M).

From the above figure, first coil carries current i_{1} and its self inductance is L_{1}. Along with its self inductance it has to face mutual induction due to rate of change of current i_{2} in the second coil.
Same case happens in the second coil also.
Dot convention is used to mark the polarity of the mutual induction.
Suppose two coils are placed nearby.

Coil 1 carries I_{1} current having N_{1} number of turn. Now the flux density created by the coil 1 is B_{1}. Coil 2 with N_{2} number of turn gets linked with this flux from coil 1. So flux linkage in coil 2 is N_{2} . φ_{21} [φ_{21} is called leakage flux in coil 2 due to coil 1].
Consider φ_{21} is also changing with respect to time, so an EMF appears across coil 2.
This EMF is called mutually induced EMF.

Now it can be written from these equations,
Again, coil 1 gets induced by flux from coil 2 due to current I2 in the coil 2.
In same manner it can be written that
for coil 1.
However, using the reciprocity theorem which combines Ampere’s law and the Biot-Savart law, one may show that the constants are equal. i.e. M_{12} = M_{21} = M.
M is the mutual inductance for both coil in Henry.
The value of mutual inductance is a function of the self-inductances
Suppose two coils are place nearby such that they are in mutual induction. L_{1} and L_{2} are co-efficient of self induction of them. M is the mutual inductance.
Here, ƙ is called co-efficient of coupling and it is defined as the ratio of mutual inductance actually present between the two coils to the maximum possible value.
If the flux due to first coil completely links with second coil, then ƙ = 1, then two coils are tightly coupled. Again if no linkage at all then ƙ = 0 and hence two coils are magnetically isolated.
Merits and demerits of mutual inductance:
Due to mutual inductance, transformer establishes its operating principle.
But due to mutual inductance, in any circuit having inductors, has to face extra voltage drop.

How to find out L_{eq} in a circuit having mutual inductance with dot convention
Suppose two coils are in series with same place dot.
**Mutual inductance** between them is positive.
Suppose two coils are in series with opposite place dot.
When a few numbers of inductors are in series with mutual inductances.