NCERT · Class 12 · Physics · Electromagnetic InductionDerive an expression for the mutual inductance of two long coaxial solenoids of same length $L$. State the factors on which the mutual inductance of two coils depends.

Mutual Inductance of Two Long Coaxial Solenoids

1. Introduction and Definition\nMutual inductance is the property of two coils due to which an induced electromotive force (emf) is produced in one coil when the magnetic flux linked with it changes as a result of a change in current in the neighbouring coil.

\nConsider two long coaxial solenoids $S_1$ and $S_2$ of equal length $L$.

• Let $N_1$ and $N_2$ be the total number of turns in solenoids $S_1$ and $S_2$ respectively.
• Let $n_1 = \frac{N_1}{L}$ and $n_2 = \frac{N_2}{L}$ be the number of turns per unit length of $S_1$ and $S_2$.
• Let $A$ be the area of cross-section of the inner solenoid $S_1$.

2. Mathematical Derivation\nLet a current $I_1$ pass through the inner solenoid $S_1$.

1. Magnetic Field produced inside $S_1$: $$B_1 = \mu_0 n_1 I_1 = \mu_0 \left(\frac{N_1}{L}\right) I_1$$

2. Magnetic Flux linked with each turn of solenoid $S_2$: $$\Phi_2 = B_1 A = \mu_0 \left(\frac{N_1}{L}\right) I_1 A$$

3. Total Magnetic Flux Linkage with Solenoid $S_2$ ($N_2$ turns): $$N_2 \Phi_2 = N_2 \left[ \mu_0 \left(\frac{N_1}{L}\right) I_1 A \right] = \frac{\mu_0 N_1 N_2 A}{L} I_1$$

4. Expression for Mutual Inductance ($M_{21}$): By definition of mutual inductance, $N_2 \Phi_2 = M_{21} I_1$ $$M_{21} = \frac{\mu_0 N_1 N_2 A}{L} = \mu_0 n_1 n_2 A L$$ \nSimilarly, if a current $I_2$ flows through solenoid $S_2$, the flux linkage with $S_1$ gives: $$M_{12} = M_{21} = M$$ \nHence, the mutual inductance between two long coaxial solenoids is: $$M = \frac{\mu_0 N_1 N_2 A}{L} = \mu_0 n_1 n_2 A L$$ \nIf a medium of relative permeability $\mu_r$ is placed inside the solenoids, then: $$M = \frac{\mu_0 \mu_r N_1 N_2 A}{L}$$

3. Factors Affecting Mutual Inductance

• Number of Turns ($N_1, N_2$): Mutual inductance is directly proportional to the product of the number of turns in both coils ($M \propto N_1 N_2$).
• Area of Cross-Section ($A$): Larger cross-sectional area increases flux linkage, thus increasing $M$.
• Permeability of Core Material ($\mu_r$): Inserting a soft iron core with high relative permeability increases the value of $M$ significantly.
• Length of Solenoids ($L$): Mutual inductance is inversely proportional to the length for fixed total turns.
• Distance and Orientation: Placing coils closer together or coaxially increases coupling efficiency and hence increases mutual inductance.