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Electromagnetic Induction

Ch. 12Std 12

Easy Overview

If electric currents produce magnetic fields, can magnetic fields produce electric currents? Michael Faraday asked this question in the 1830s, and the answer changed the world. Yes — a changing magnetic field creates an electric field, which can drive a current in a conductor. This is electromagnetic induction. Faraday's law states that the induced EMF in a coil equals the negative rate of change of magnetic flux: ε = −dΦ/dt. The minus sign is Lenz's law — the induced current flows in a direction that opposes the change causing it. It is nature's way of conserving energy. If induced current helped the change, you would get infinite energy for free — impossible. Magnetic flux Φ = BA cosθ. The flux through a coil can change because: (1) the magnetic field B changes, (2) the area A of the loop changes, (3) the angle θ changes, or (4) any combination. This is the principle behind electric generators — rotating a coil in a magnetic field changes flux sinusoidally, producing alternating current. Self-induction occurs when a changing current in a coil induces an EMF in the same coil. The back EMF is ε = −L dI/dt, where L is self-inductance. A coil with high L opposes rapid changes in current — it acts as a 'current flywheel'. Mutual induction occurs between two coils, and is the principle of the transformer, which steps AC voltages up or down. Induction also explains eddy currents — circulating currents induced in bulk conductors, used in induction heating and braking.

Magnetic Flux

Φ = B·A = BA cosθ. Unit: weber (Wb). 1 Wb = 1 T·m². For a coil of N turns: Φ_total = NΦ. Flux through a surface depends on orientation — maximum when area perpendicular to B, zero when parallel.

Faraday's Law of Induction

ε = −dΦ/dt for a single loop. For N turns: ε = −N dΦ/dt. The induced EMF is proportional to the rate of change of flux, not the flux itself. A static magnetic field produces zero EMF. This is what makes AC generators work — the coil rotates, so flux changes continuously.

Lenz's Law

The induced current always opposes the change that caused it. If you push a magnet into a coil, the induced current creates a magnetic field that pushes back. If you pull it out, the induced current pulls it back. This is why you feel resistance when moving a magnet near a conductor. Lenz's law ensures conservation of energy — work must be done to generate electrical energy.

Motional EMF

A conductor of length L moving perpendicular to B with speed v: ε = BLv. Derived from Lorentz force: F = qvB on electrons. Work done by magnetic force = qvBL. Equivalent to rate of cutting flux lines. For a rod sliding on rails: ε = BLv regardless of path shape.

AC Generator (Alternator)

A rectangular coil rotates in uniform B with angular speed ω. Flux Φ = NBA cosωt. EMF ε = −dΦ/dt = NBAω sinωt = ε₀ sinωt. Peak EMF ε₀ = NBAω. Frequency: f = ω/(2π). India: 50 Hz. Carbon brushes and slip rings deliver AC. Simple and robust.

Eddy Currents

Changing B induces circulating currents in bulk conductors. Effects: (1) heating (useful in induction cooktops), (2) braking (magnetic brakes in trains), (3) damping. Wasteful in transformers — laminated cores break eddy current paths and reduce losses. Also used in metal detectors.

Self-Inductance

For a coil: Φ = LI. ε_L = −L dI/dt. L = NΦ/I. Solenoid: L = μ₀N²A/l. Unit: henry (H). 1 H = 1 V·s/A. A 1 H coil produces 1 V back EMF when current changes at 1 A/s. Inductors oppose rapid current changes — they smooth current.

Energy Stored in an Inductor

U = ½LI². Energy is stored in the magnetic field. Energy density: u = B²/(2μ₀). The work done to build up current is stored as magnetic energy. When the circuit opens, this energy dissipates — sometimes as a spark across the switch.

Mutual Inductance

Two coils: current I₁ in coil 1 creates flux through coil 2: Φ₂ = MI₁. ε₂ = −M dI₁/dt. M = mutual inductance. M = k√(L₁L₂) where k ≤ 1 is coupling coefficient. For coaxial solenoids: M = μ₀N₁N₂A/l.

Transformer

Alternating current in primary creates changing flux. Mutual induction induces EMF in secondary. V_s/V_p = N_s/N_p. For an ideal transformer: V_pI_p = V_sI_s. Step-up: N_s > N_p (high V, low I). Step-down: N_s < N_p (low V, high I). Efficiency can exceed 99% in modern transformers.

Power Transmission

Power loss in transmission lines: P_loss = I²R. To reduce loss: transmit at high voltage (low current). Generators produce ~20 kV, stepped up to 400 kV or more for transmission, then stepped down for distribution. This is why AC (which can be transformed easily) dominates power systems.

Induction Cooktop

A coil below the glass top carries high-frequency AC (20-100 kHz). The changing magnetic field induces eddy currents in the ferromagnetic cookware. The eddy currents heat the pan directly via I²R heating. The glass top stays cool — only the pan gets hot. Fast and efficient.

Key Points

  • •Magnetic flux Φ = BA cosθ. Unit: weber (Wb).
  • •Faraday's law: ε = −N dΦ/dt.
  • •Lenz's law: induced current opposes the change.
  • •Motional EMF: ε = BLv.
  • •AC generator: ε = NBAω sinωt.
  • •Eddy currents: circulating currents in bulk conductors.
  • •Self-inductance: L = NΦ/I. ε = −L dI/dt. Energy: ½LI².
  • •Mutual inductance: M = k√(L₁L₂). V_s/V_p = N_s/N_p.
  • •Transformer: step-up/step-down AC voltages.

Practice Questions

  • State Faraday's law of electromagnetic induction. Derive expression for induced EMF in a coil rotating in uniform magnetic field (AC generator).
  • State Lenz's law. Explain with examples why it follows the law of conservation of energy.
  • What is motional EMF? Derive ε = BLv for a conductor moving perpendicular to B. A 0.5 m rod moves at 10 m/s in 0.2 T field. Find induced EMF.
  • Explain self-inductance. Derive L for a solenoid. A solenoid 0.5 m long, 2 cm² area, 1000 turns. Find L. Energy stored at 2 A?
  • Explain mutual inductance. A transformer with 200 primary turns, 1000 secondary turns. Primary voltage 230 V. Find secondary voltage. If secondary current is 1 A, find primary current (ideal).
  • What are eddy currents? State two useful applications and two ways to minimise losses.
  • Derive expression for energy stored in an inductor. Show energy density = B²/(2μ₀).
  • Explain high-voltage power transmission. Why is AC preferred over DC for long-distance transmission?