Question

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

• A. $L_1+L_2$
• B. $\frac{1}{2}(L_1+L_2)$
• C. $(L_1\pm L_2)$
• D. $\sqrt{L_1L_2}$

Answer 1

Answer: (D)

$\sqrt{L_1L_2}$

Answer 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 $M _{12}=\frac{ N _{2} \phi_{ B _{2}}}{ i _{1}}$ and $M _{21}=\frac{ N _{1} \phi_{ B _{1}}}{ i _{2}}$ and $L _{1}=\frac{ N _{1} \phi_{ B _{1}}}{ i _{1}}$ and $L _{2}=\frac{ N _{2} \phi_{ B _{2}}}{ i _{2}}$ If all the flux of coil $2$ links coil $1$ and vice versa, then $\phi_{ B _{2}}=\phi_{ B _{1}}$ As $M _{12}= M _{21}= M$ Thus we get $M _{12} M _{21}= M ^{2}=\frac{ N _{1} N _{2} \phi_{ B _{1}} \phi_{ B _{2}}}{ i _{1} i _{2}}= L _{1} L _{2}$ 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.