CBSE · Class 12 · Physics · Electromagnetic InductionWhat do you mean by mutual induction? Derive an expression for the mutual inductance ($M$) of two long coaxial solenoids of length $L$, cross-sectional area $A$, having $N1$ and $N2$ turns respectively. Also, list the factors on which mutual inductance depends.

Mutual Induction\nMutual induction is the phenomenon in which an electromotive force (emf) is induced in a coil (secondary coil) whenever there is a change in electric current flowing through a neighboring coil (primary coil).

Derivation of Mutual Inductance of Two Coaxial Solenoids

\nConsider two long coaxial solenoids $S_1$ and $S_2$ of equal length $L$ and cross-sectional area $A$.

• Let $N_1$ = total number of turns in inner solenoid $S_1$
• Let $N_2$ = total number of turns in outer solenoid $S_2$
• Number of turns per unit length in $S_1$: $n_1 = \frac{N_1}{L}$
• Number of turns per unit length in $S_2$: $n_2 = \frac{N_2}{L}$

Step 1: Magnetic field due to current in $S_1$\nWhen a current $I_1$ flows through solenoid $S_1$, a uniform magnetic field $B_1$ is produced inside it along its axis:

$$ B_1 = \mu_0 n_1 I_1 = \mu_0 \left(\frac{N_1}{L}\right) I_1 $$

Step 2: Magnetic flux linked with one turn of $S_2$\nSince solenoid $S_2$ completely encloses or is coaxial with $S_1$, the magnetic flux linked with each single turn of $S_2$ due to field $B_1$ is:

$$ \Phi_2 = B_1 A = \left(\frac{\mu_0 N_1 I_1}{L}\right) A $$

Step 3: Total magnetic flux linked with solenoid $S_2$\nTotal flux linkage in $S_2$ consisting of $N_2$ turns is:

$$ N_2 \Phi_2 = N_2 \left[ \frac{\mu_0 N_1 I_1 A}{L} \right] = \frac{\mu_0 N_1 N_2 A}{L} I_1 $$

Step 4: Formula for Mutual Inductance ($M_{21}$)\nBy definition of mutual inductance:

$$ N_2 \Phi_2 = M_{21} I_1 $$\nEquating both expressions: $$ M_{21} I_1 = \frac{\mu_0 N_1 N_2 A}{L} I_1 $$ $$ M = \frac{\mu_0 N_1 N_2 A}{L} $$\nAlternatively, in terms of turns per unit length ($n_1$ and $n_2$): $$ M = \mu_0 n_1 n_2 A L $$

Factors Affecting Mutual Inductance

1. Number of Turns ($N_1, N_2$): $M \propto N_1 N_2$. Mutual inductance increases with more turns in either coil.
2. Area of Cross-section ($A$): $M \propto A$. Larger area leads to higher mutual inductance.
3. Permeability of Core Material ($\mu_r$): If a core of relative permeability $\mu_r$ (like soft iron) is inserted, $M = \frac{\mu_0 \mu_r N_1 N_2 A}{L}$.
4. Relative Orientation and Separation: Mutual inductance is maximum when coils are coaxially wound over each other and decreases as separation increases.