MP Board · Class 12 · Physics · Electromagnetic Induction(a) What is meant by Mutual Induction? Define the Coefficient of Mutual Inductance and state its S.I. unit and dimensions. (b) Derive an expression for the mutual inductance ($M$) of two long coaxial solenoids $S1$ and $S2$ of equal length $l$, where solenoid $S1$ has $N1$ turns and radius $r1$, and solenoid $S2$ has $N2$ turns and radius $r2$ (with $r1

Mutual Induction

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

Coefficient of Mutual Inductance ($M$)

• If $I_1$ is the current flowing in the primary coil and $\Phi_2$ is the magnetic flux linked with each turn of the secondary coil having $N_2$ turns, then total flux linkage is proportional to $I_1$: $$N_2 \Phi_2 \propto I_1 \implies N_2 \Phi_2 = M I_1$$ where $M$ is the Coefficient of Mutual Inductance (or Mutual Inductance).
• Definition: Mutual inductance is defined as the total magnetic flux linked with the secondary coil when a unit current flows through the primary coil.
• Alternative Definition: According to Faraday's Law, induced emf in the secondary coil is: $$\mathcal{E}_2 = -M \frac{dI_1}{dt}$$ Thus, mutual inductance is numerically equal to the emf induced in the secondary coil when the rate of change of current in the primary coil is unity ($1\text{ A/s}$).
• S.I. Unit: Henry (H) or $\text{V}\cdot\text{s}/\text{A}$ or $\text{Wb}/\text{A}$.
• Dimensional Formula: $[M^1 L^2 T^{-2} A^{-2}]$.

Derivation for Mutual Inductance of Two Coaxial Solenoids

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

• Inner solenoid $S_1$: Radius $r_1$, total turns $N_1$, turns per unit length $n_1 = N_1 / l$.
• Outer solenoid $S_2$: Radius $r_2$, total turns $N_2$, turns per unit length $n_2 = N_2 / l$. \nSuppose a current $I_2$ flows through the outer solenoid $S_2$:

1. Magnetic field inside solenoid $S_2$ is: $$B_2 = \mu_0 n_2 I_2 = \mu_0 \left(\frac{N_2}{l}\right) I_2$$
2. The magnetic flux linked with a single turn of the inner solenoid $S_1$ (area $A_1 = \pi r_1^2$) due to $B_2$ is: $$\Phi_1 = B_2 A_1 = \left(\mu_0 \frac{N_2}{l} I_2\right) (\pi r_1^2)$$
3. Total magnetic flux linked with all $N_1$ turns of solenoid $S_1$: $$N_1 \Phi_1 = N_1 \left( \mu_0 \frac{N_2}{l} I_2 \pi r_1^2 \right) = \frac{\mu_0 N_1 N_2 \pi r_1^2}{l} I_2$$
4. By definition of mutual inductance $N_1 \Phi_1 = M_{12} I_2$, we get: $$M_{12} = \frac{\mu_0 N_1 N_2 \pi r_1^2}{l}$$
5. By reciprocity theorem, $M_{12} = M_{21} = M$, hence: $$M = \frac{\mu_0 N_1 N_2 A_1}{l} = \mu_0 n_1 n_2 A_1 l$$

Factors Affecting Mutual Inductance

1. Number of Turns: $M$ is directly proportional to the product of number of turns in both coils ($M \propto N_1 N_2$).
2. Relative Orientation & Separation: $M$ is maximum when coils are coaxial/close to each other and decreases as the distance between them increases or orientation changes.