Question

Two different coils of self inductance ${L_1}$ and ${L_2}$ are placed close to each other so that the effective flux in one coil is completely linked with the other. If $M$ is the mutual inductance between them, then
(A) $M = \dfrac{{{L_1}}}{{{L_2}}}$
(B) $M = {L_1}{L_2}$
(C) $M = \sqrt {{L_1}{L_2}}$
(D) $M = {L_1}{L_2}$

Answer 1

Hint : The self inductance of a coil is measured in Henry. The mutual inductance ${L_1}$ and ${L_2}$ of both the coils will be dependent on the distance between them if we have to calculate how much magnetic flux lines pass through each but since it is written that they are very near to each other we will the constant is $1$ . Then next we will find the value of M using both the values of self inductances given to us and then multiply both the value and square root the value of ${M^2}$ obtained after multiplying both the value of $M$ .

Complete Step By Step Answer:
The value of $M$ for both coils can be written as:
$M = - \frac{{{e_2}}}{{d{i_1}/dt}} = - \frac{{{e_1}}}{{d{i_2}/dt}}$
Where ${e_1}$ and ${e_2}$ are the induced emf
Upon putting the value ${e_1}$ and ${e_2}$ as
${e_1} = - {L_1}\frac{{d{i_1}}}{{dt}}{\text{ and }}{e_2} = - {L_2}\frac{{d{i_2}}}{{dt}}$
Where ${L_1}$ and ${L_2}$ are mutual inductance of both coils we then write after multiplying both the values of $M$ as
${M^2} = \dfrac{{{e_1}{e_2}}}{{\left( {\dfrac{{d{i_1}}}{{dt}}} \right)\left( {\dfrac{{d{i_2}}}{{dt}}} \right)}}$
Substituting back the value of ${L_1}$ and ${L_2}$ and then finding the square root of the equation we get
$M^2 = L_1L_2 \Rightarrow M = L_1L_2 - - - - \surd$ ${M^2} = {L_1}{L_2} \Rightarrow M = \sqrt {{L_1}{L_2}}$
Thus the correct option of the given question will be $C$ .

Note :
The mutual inductance is a change of emf in another coil due to change in current in the neighbouring coil. This phenomenon means that one coil tries to oppose the change in the condition of flow of current in another coil.