Question

How does mutual inductance of a pair of coil changes when
(i) Distance between the coils is increased
(ii) Number of turns in both the coils is increased

Answer 1

Hint: Mutual induction works based on the concept of magnetic flux. We’ll find out the relation between magnetic flux and distance in terms of electromagnetic induction. Also we have to look into how the number of turns decides the strength of mutual induction.

Complete step by step answer:
To understand how mutual induction works, let’s first have a look at its concept. Mutual induction is the phenomenon of production of an induced electromotive force or electric current in a coil as a result of changing current in a neighbouring coil placed in close proximity.
When current flows through the primary coil, magnetic flux linked with the primary coil also increases. Since the secondary coil is placed in close proximity of the primary coil, magnetic flux linked with the secondary conductor also increases.
Faraday’s first law of electromagnetic induction says, as long as there is change in magnetic flux linked with a coil, an electro-motive force or current is induced in the coil.
So, magnetic flux linked with the secondary coil $(\varphi )$ is directly proportional to current flowing through the first coil or
$\varphi \propto I$
$\Rightarrow \varphi =MI$
Here $M$ is a proportionality constant known as coefficient of mutual induction
At constant current, $M=\varphi$ ………. (1)
We know that magnetic flux is inversely proportional to distance. The more is the distance, the less This answers (i), as the distance between the coils increases, the mutual inductance will decrease.
Let’s now have a look at how the number of turns in both the coils affect mutual induction.
Mutual inductance between two coils can be found out by using the formula
$M={\mu }_0{N}_1{N}_{2}AI$
Where, $M$ is the mutual inductance
${\mu }_0$ is permeability of air
$N_1$ is number of turns in the primary coil
$N_2$ is number of turns in the secondary coil
$A$ is the area of any face of the coil
$I$ is the current flowing through the primary coil
In the problem given, the current and area of the faces of the coil are constant.
So from this data we can conclude that mutual inductance is directly proportional to the number of turns in any coil.
That is $M\propto N_1$ and $M\propto N_2$
From this we can find our answer to (ii), the value of mutual inductance will increase if the number of turns in any or both of the coils is increased.

Note: The mutual inductance of a coil is measured in Henry $(H)$. One Henry will be the mutual inductance of the secondary coil if an electromotive force of $1V$ is induced by a current changing at the rate of $1A{s}^{-1}$ in the primary coil. Dimensionally self-induction and mutual induction are the same but they vary in their concepts. Sometimes we are required to calculate mutual inductance from self-inductance. If self-inductance is given for two coils then there collective mutual inductance can be calculated by the formula $M=\sqrt{L_1{L}_2}$ where $L_1$ and $L_2$ are self-inductances of primary and secondary coils respectively.