Question

Two coils of self inductances $L_1$ and $L_2$ are placed so close together that effective flux in one coil is completely linked with the other. If M is the mutual inductance between them, then

• A. $M=L_1{L}_2$
• B. $M=L_1/L_2$
• C. $M=(L_1{L}_2{)}^2$
• D. $M=\sqrt{\left(L_1{L}_2\right)}$

Answer 1

The correct option is D $M=\sqrt{\left(L_1{L}_2\right)}$

Consider two coils 1 and 2

When current $I_1$ flows through coil 1 , $L_1=\frac{N_1{\varphi }_1}{I_1}$ and $M_{12}=M=\frac{N_2{\varphi }_{12}}{I_1}$

where ${\varphi }_{12}=k{\varphi }_1$

(Note : when current flows through one coil it produces flux; the whole flux may not link with the other coil coupled and this is due to leakage flux by a fraction k, known as coefficient of coupling)

When current $I_2$ flows through coil 2 , $L_2=\frac{N_2{\varphi }_2}{I_2}$ and $M_{21}=M=\frac{N_1{\varphi }_{21}}{I_2}$

where ${\varphi }_{21}=k{\varphi }_2$

Using above equations, $M^2=\frac{N_{2}k{\varphi }_1}{I_1}\times \frac{N_{1}k{\varphi }_2}{I_2}=k^2{L}_1{L}_2$

As the coils are completely link with each other so it is is called magnetically tightly coupled and so $k=1$.

Thus, $M^2=L_1{L}_2$ or $M=\sqrt{L_1{L}_2}$