Chapter 6: Electromagnetic Induction

ELECTROMAGNETIC INDUCTION

Faraday's Experiments

  • Discovery: Change in magnetic flux through a circuit induces EMF
  • Demonstration: Moving magnet near coil generates current
  • Key Insight: Relative motion between magnet and coil is what matters

Magnetic Flux

\[ \Phi_m = \vec{B} \cdot \vec{A} = BA\cos\theta \]
  • Unit: Weber (Wb) = Tesla·m² (T·m²)
  • Physical Meaning: Number of magnetic field lines passing through a surface

FARADAY'S LAW OF INDUCTION

Statement

Induced EMF in a circuit equals negative rate of change of magnetic flux through the circuit

\[ \varepsilon = -\frac{d\Phi_m}{dt} \]

Lenz's Law

  • Statement: Induced current flows in direction that opposes the change causing it
  • Physical Basis: Conservation of energy
  • Application: Determines direction of induced EMF and current

Methods of Changing Magnetic Flux

  1. Changing magnetic field (B)
  2. Changing area (A)
  3. Changing angle (θ) between B and A

MOTIONAL ELECTROMOTIVE FORCE

Conductor Moving in Magnetic Field

\[ \varepsilon = Blv \]
  • where l = length of conductor, v = velocity perpendicular to B
  • Force on Charges: F = qvB
  • Direction: Determined by right-hand rule

Eddy Currents

  • Definition: Circulating currents induced in bulk conductors
  • Causes: Changing magnetic flux through conductor
  • Effects: Heating, magnetic damping
  • Applications: Induction heating, electromagnetic brakes
  • Reduction: Laminating the conductor (e.g., transformer cores)

ENERGY CONSIDERATIONS

Work Done Against Induced EMF

\[ W = \int \varepsilon \cdot i \cdot dt \]

Mechanical energy converted to electrical energy

Mechanical Energy and Electrical Energy

  • Conservation of energy principle applies
  • Mechanical work done = Electrical energy produced + Heat losses

INDUCTANCE

Self-Inductance

  • Definition: Flux linkage per unit current through the same circuit
  • Mathematical Expression: L = NΦ/I
  • Unit: Henry (H) = Weber/Ampere (Wb/A)
  • Physical Meaning: Opposition to change in current

Self-Induced EMF

\[ \varepsilon = -L\frac{dI}{dt} \]

Opposes change in current

Inductance of a Solenoid

\[ L = \mu_0 n^2 Al \]
  • n = number of turns per unit length
  • A = cross-sectional area
  • l = length of solenoid

Mutual Inductance

  • Definition: Flux linkage in one circuit per unit current in another circuit
  • Mathematical Expression: M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂
  • Unit: Henry (H)
  • Symmetry: M₁₂ = M₂₁

Mutual Induced EMF

\[ \varepsilon_2 = -M\frac{dI_1}{dt} \]
\[ \varepsilon_1 = -M\frac{dI_2}{dt} \]

ENERGY STORED IN INDUCTOR

Energy Formula

\[ U = \frac{1}{2}LI^2 \]

Energy stored in magnetic field

Energy Density

\[ u = \frac{1}{2}\frac{B^2}{\mu_0} \]

Energy per unit volume in magnetic field

AC GENERATOR

Working Principle

  • Faraday's Law: Rotating coil in magnetic field induces EMF
  • EMF: ε = NABω·sin(ωt)
  • Frequency: f = ω/2π = rotation frequency

Components

  • Armature: Rotating coil
  • Field Magnet: Provides magnetic field
  • Slip Rings & Brushes: Transfer current from rotating to stationary part

TRANSFORMERS

Working Principle

  • Mutual Induction: Changing current in primary induces EMF in secondary
  • Flux Linkage: Iron core enhances flux linkage between coils

Transformer Equation

\[ \frac{V_2}{V_1} = \frac{N_2}{N_1} = \frac{I_1}{I_2} \]
  • N₁, N₂ = number of turns in primary and secondary
  • V₁, V₂ = voltages across primary and secondary
  • I₁, I₂ = currents in primary and secondary

Types

  • Step-up: V₂ > V₁ (N₂ > N₁)
  • Step-down: V₂ < V₁ (N₂ < N₁)

Efficiency

\[ \eta = \frac{P_2}{P_1} \times 100\% \]
  • P₁ = input power, P₂ = output power
  • Losses: Copper losses (I²R), Iron losses (hysteresis, eddy currents)

APPLICATIONS

Electromagnetic Damping

  • Principle: Eddy currents oppose motion
  • Applications: Galvanometer damping, electromagnetic brakes

Induction Furnace

  • Principle: Eddy currents heat conducting material
  • Advantage: Non-contact heating

Electromagnetic Flowmeter

  • Principle: Moving conductor (fluid) in magnetic field generates EMF
  • Application: Measuring flow rate of conducting fluids

KEY FORMULAS

  • Magnetic Flux: Φₘ = B·A·cosθ
  • Faraday's Law: ε = -dΦₘ/dt
  • Motional EMF: ε = Blv
  • Self-Inductance: L = NΦ/I
  • Self-Induced EMF: ε = -L·dI/dt
  • Mutual Induced EMF: ε₂ = -M·dI₁/dt
  • Energy in Inductor: U = (1/2)LI²
  • Energy Density: u = (1/2)B²/μ₀
  • AC Generator EMF: ε = NABω·sin(ωt)
  • Transformer Equation: V₂/V₁ = N₂/N₁ = I₁/I₂