Question

Two coils of inductances $L_1$ , and $L_2$ are linked such that their mutual inductance is $M$. Then $M=$

• A. $L_1+L_2$
• B. $\frac{1}{2}(L_1+L_2)$
• C. $(L_1\pm L_2)$
• D. $\sqrt{L_1{L}_2}$

Answer 1

The correct option is D $\sqrt{L_1{L}_2}$

Let us first consider a case when the total flux associated with one coil links with the other, i.e. a case of maximum flux linkage. Consider two coils placed adjacent to each other. Thus

$M_{12}=\frac{N_2{\varphi }_{B_2}}{i_1}$ and $M_{21}=\frac{N_1{\varphi }_{B_1}}{i_2}$

and

$L_1=\frac{N_1{\varphi }_{B_1}}{i_1}$ and $L_2=\frac{N_2{\varphi }_{B_2}}{i_2}$

If all the flux of coil 2 links coil 1 and vice versa, then

${\varphi }_{B_2}={\varphi }_{B_1}$

As $M_{12}=M_{21}=M$

Thus we get

$M_{12}{M}_{21}=M^2=\frac{N_1{N}_2{\varphi }_{B_1}{\varphi }_{B_2}}{i_1{i}_2}=L_1{L}_2$

or

$M=\sqrt{L_1{L}_2}$(assuming that there is no flux leakage)

Mutual inductance is inductance of emf in a coil due to change in current in another case

for two coils which are mutually coupled has mutual inductance in $=\sqrt{L_1{L}_2}$

This is a general result assuming that there is no flux leakage.