Mutual Inductance — Revision Notes

⚡ 30-Second Revision

Definition — Changing current in one coil induces EMF in a nearby coil.
Mutual Inductance (M) — $M = \frac{\Phi_2}{I_1}$ (or $\frac{\Phi_1}{I_2}$), SI unit: Henry (H).
Induced EMF — $E_2 = -M \frac{dI_1}{dt}$.
Factors affecting M — Number of turns ($N_1, N_2$), geometry, relative orientation, distance, core material ($\mu$).
Coefficient of Coupling (k) — $M = k \sqrt{L_1 L_2}$, where $0 \le k \le 1$.
Lenz's Law — Induced EMF opposes the change in current.

2-Minute Revision

Mutual inductance (M) describes the magnetic coupling between two separate coils. When the current in one coil (primary) changes, it produces a changing magnetic field, which in turn causes a changing magnetic flux through the nearby second coil (secondary).

According to Faraday's Law, this changing flux induces an electromotive force (EMF) in the secondary coil. The magnitude of this induced EMF is directly proportional to the rate of change of current in the primary coil, with M as the constant of proportionality: $E_2 = -M \frac{dI_1}{dt}$.

The negative sign signifies Lenz's Law, meaning the induced EMF opposes the change in current. Mutual inductance depends on the number of turns in both coils, their sizes, shapes, relative orientation, distance, and the magnetic permeability of the medium between them.

It is independent of the current itself. The coefficient of coupling ($k$) quantifies the degree of magnetic linkage, relating M to the self-inductances ($L_1, L_2$) as $M = k \sqrt{L_1 L_2}$. Key applications include transformers and wireless charging.

5-Minute Revision

Mutual inductance is a crucial concept in electromagnetic induction, explaining how two distinct circuits can interact magnetically. When a current $I_1$ flows through a primary coil, it generates a magnetic field.

If a secondary coil is placed nearby, some of this magnetic field passes through it, creating a magnetic flux linkage, $\Phi_2$. The mutual inductance $M$ is defined as the ratio of this flux linkage to the current causing it: $M = \frac{\Phi_2}{I_1}$.

The SI unit for mutual inductance is the Henry (H). If the current $I_1$ in the primary coil changes, the magnetic flux $\Phi_2$ linked with the secondary coil also changes. According to Faraday's Law, this changing flux induces an EMF $E_2$ in the secondary coil, given by $E_2 = -M \frac{dI_1}{dt}$.

The negative sign is a consequence of Lenz's Law, indicating that the induced EMF (and resulting current) will oppose the change in the primary current.

The value of mutual inductance $M$ is not constant for all coil arrangements; it depends on several factors: the number of turns in each coil ($N_1, N_2$), their geometric shapes and sizes, their relative orientation (e.

g., parallel, perpendicular), the distance separating them, and the magnetic permeability of the core material (e.g., air, iron). For instance, for two coaxial solenoids, $M = \frac{\mu_0 N_1 N_2 A}{l}$.

The degree of magnetic coupling between two coils is quantified by the dimensionless coefficient of coupling, $k$, where $0 \le k \le 1$. It relates mutual inductance to the individual self-inductances $L_1$ and $L_2$ of the coils: $M = k \sqrt{L_1 L_2}$.

A value of $k=1$ signifies perfect coupling, as seen in ideal transformers. Understanding these principles is vital for solving numerical problems and conceptual questions related to transformers, induction heating, and other applications of electromagnetic induction.

Prelims Revision Notes

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Definition — Mutual inductance (M) is the property of two coils where a changing current in one coil induces an EMF in the other. It quantifies magnetic coupling.
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Formula for M — $M = \frac{\Phi_2}{I_1}$ (flux in coil 2 due to current in coil 1). $M$ is symmetrical, so $M_{12} = M_{21}$.
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SI Unit — Henry (H). $1,\text{H} = 1,\text{Wb/A} = 1,\text{V s/A}$.
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Induced EMF — $E_2 = -M \frac{dI_1}{dt}$. The negative sign is from Lenz's Law, indicating opposition to the change in current.
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Factors Affecting M — M is a geometric and material constant for a given setup.

* Number of turns: $M \propto N_1 N_2$. * Geometry: Size, shape of coils. * Relative Orientation: Max when parallel, min (zero) when perpendicular. * Distance: M decreases rapidly with increasing distance. * Core Material: $M \propto \mu$ (permeability of the medium). Ferromagnetic cores increase M significantly.

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Coefficient of Coupling (k) — A dimensionless quantity ($0 \le k \le 1$) representing the fraction of flux from one coil linking with the other.

* Formula: $M = k \sqrt{L_1 L_2}$, where $L_1, L_2$ are self-inductances. * $k=1$ for perfect coupling (e.g., ideal transformer). * $k=0$ for no coupling.

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Example Derivation (Coaxial Solenoids) — For a long solenoid (length $l$, area $A$, $N_1$ turns) with a smaller coil ($N_2$ turns) wound around its center, $M = \frac{\mu_0 N_1 N_2 A}{l}$.
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Applications — Transformers, induction cooktops, wireless chargers, RFID.
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Key Distinction — Self-inductance (L) is for a single coil and its own current; Mutual inductance (M) is between two coils and their interacting currents.
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Problem Solving — For numericals, ensure correct units. For $I(t)$ varying sinusoidally, remember differentiation: $\frac{d}{dt}(\sin(\omega t)) = \omega \cos(\omega t)$. Pay attention to the magnitude vs. direction of EMF.

Vyyuha Quick Recall

To remember factors affecting Mutual Inductance (M): Nice Girls Often Dance Carefully

N — Number of turns
G — Geometry (size, shape)
O — Orientation (relative)
D — Distance between coils
C — Core material (permeability)