EMI (Electromagnetic Induction)

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

EMI and AC — Key Facts

Faraday’s Law of Electromagnetic Induction:

Induced EMF equals the negative rate of change of magnetic flux: $$\varepsilon = -\frac{d\Phi}{dt}$$

The minus sign indicates Lenz’s Law — induced current opposes the change in flux.

Magnetic Flux: $$\Phi = \vec{B} \cdot \vec{A} = BA\cos\theta$$

Unit: Weber (Wb), where 1 Wb = 1 T·m²

Lenz’s Law: The direction of induced current is such that it opposes the change that produced it.

⚡ JEE Exam Tip: Always apply Lenz’s Law before determining current direction. If flux is increasing, induced B is opposite to external B. If flux is decreasing, induced B is in same direction as external B.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Motional EMF:

When a conductor of length ℓ moves with velocity v perpendicular to magnetic field B: $$\varepsilon = B\ell v$$

General case (at angle θ to perpendicular): $\varepsilon = B\ell v\sin\theta$

This is equivalent to the rate of change of flux through the moving loop.

Self-Induction:

When current in a coil changes, changing flux through the coil induces EMF in itself: $$\varepsilon = -L\frac{dI}{dt}$$

Self-inductance L depends on coil geometry:

• Solenoid: $L = \mu_0 n^2 A \ell$ (n = turns per unit length, A = area, ℓ = length)
• Toroid: $L = \mu_0 n^2 \cdot 2\pi r \cdot A$

Energy stored in inductor: $$U = \frac{1}{2}LI^2$$

Mutual Induction:

When current in one coil changes, it induces EMF in a nearby coil: $$\varepsilon_2 = -M\frac{dI_1}{dt}$$

Mutual inductance M depends on geometry of both coils and their separation.

For two coils with N₁ and N₂ turns: $M = k\sqrt{L_1 L_2}$ where k = coefficient of coupling (0 ≤ k ≤ 1)

⚡ JEE Exam Tip: If two inductors with self-inductances L₁ and L₂ have mutual inductance M, the equivalent inductance depends on whether they’re connected in series (with or without coupling) or parallel. For series with aiding coupling: $L_{eq} = L_1 + L_2 + 2M$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Induced Electric Field:

Changing magnetic field produces a circulating (non-conservative) electric field: $$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$$

This is Faraday’s Law in integral form — fundamentally different from electrostatics where $\oint \vec{E} \cdot d\vec{l} = 0$.

AC Generator:

Converts mechanical energy to electrical energy using electromagnetic induction.

For N-turn coil rotating at angular velocity ω in uniform B: $$\varepsilon = \varepsilon_0 \sin(\omega t)$$

where $\varepsilon_0 = NBA\omega$ (peak emf)

A = area of coil, B = magnetic field strength

Frequency: $f = \omega/2\pi$ Hz

Transformer:

Works on mutual induction principle.

Ideal transformer (100% efficient): $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

In practice:

• Copper losses (I²R in windings)
• Iron losses (hysteresis + eddy currents in core)
• Flux leakage

Efficiency: $\eta = \frac{P_{out}}{P_{in}} \approx 90-99%$ for large transformers

LR Circuit:

When switch is closed: $$I = \frac{\varepsilon}{R}(1 - e^{-t/\tau}) \quad \text{where } \tau = L/R$$

Time constant τ = L/R (time to reach 63% of final current)

LC Oscillations:

Without resistance, energy oscillates between capacitor and inductor: $$\omega_0 = \frac{1}{\sqrt{LC}}$$

Analogy with spring-mass system:

• Capacitor ↔ spring (stores potential energy)
• Inductor ↔ mass (stores kinetic energy)

Total energy: $U = \frac{1}{2}\frac{Q_0^2}{C} = \frac{1}{2}LI_0^2$ (conserved)

Eddy Currents:

Induced currents in bulk conductors when exposed to changing magnetic fields.

Applications:

• Electromagnetic braking
• Induction heating
• Eddy current damping

To minimise eddy currents: use laminated cores (thin insulated sheets), slots, or use insulators.

⚡ JEE Advanced 2023 Analysis: AC generators, transformers, and LC oscillations appeared in recent papers. For AC generator, average power output over complete cycle is zero unless load is connected — the mechanical power input equals electrical power delivered to load.

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