Faraday's Law of Electromagnetic Induction and Lenz's Law

Overview

• Faraday's Law describes how a change in magnetic field within a closed loop induces an electromotive force (emf).
• Lenz's Law states the direction of induced current will oppose the change in magnetic flux that caused it.

Faraday's Law

Setup and Basic Principles

• Coil and Magnet Experiment: Moving a magnet into a coil generates current in the coil.
  • Moving magnet into coil: current flows counterclockwise.
  • Moving magnet away from coil: current flows clockwise.
  • Speed affects current: slower movement generates smaller current, faster movement generates larger current.
  • No movement: no induced current.

Ways to Induce Current

• Changing Coil Area: Stretching or bending the coil induces current.
• Changing Angle: Rotating the coil relative to the magnetic field induces current.

Magnetic Flux Equation

• Magnetic Flux (Φ): Φ = B * A * cos(θ)
  • B: Magnetic field (Tesla, T)
  • A: Area (m²)
  • θ: Angle between magnetic field and normal to the surface
  • Unit: Weber (Wb), 1 Wb = 1 T * m²

Induced EMF and Current

• Induced EMF (ε): proportional to the rate of change of magnetic flux.
• Equation: ε = -N * (ΔΦ / Δt)
  • N: number of loops
  • ΔΦ: change in magnetic flux
  • Δt: change in time

Lenz’s Law

Principles

• Opposition to Change: Induced current opposes the change in magnetic flux that causes it.
  • Increasing flux: induced current creates a magnetic field to oppose increase.
  • Decreasing flux: induced current creates a magnetic field to support flux.

Right-Hand Rule

• Direction of Magnetic Field: Thumb points in direction of current, fingers curl in direction of magnetic field.
• Application: Helps determine direction of induced current.

Examples

• Bar Magnet and Coil: Determine induced current direction using Lenz’s Law and right-hand rule.

Calculations

• Change in Flux: ΔΦ = B * ΔA (area change) or B * A * cos(θ) (angle change)
• EMF Calculations: ε = -N * (ΔΦ / Δt)

Transformers

Basic Concepts

• Types: Step-up (increases voltage), Step-down (decreases voltage).
• Equations:
  • Voltage: (Vs/Vp) = (Ns/Np)
  • Current: (Ip/Is) = (Ns/Np)
  • Power: Vs * Is = Vp * Ip (if 100% efficient)

Efficiency

• Efficiency (%): (Output Power / Input Power) * 100
  • Most transformers are ~99% efficient.

Examples

• Worked problems on calculating voltage, current, and power in transformers.

Inductance

Concepts

• Inductor: A coil that induces emf when current changes.
• Equations:
  • Induced EMF (ε): = -L * (ΔI / Δt)
  • Inductance (L): = (μ₀ * N² * A) / l
    • μ₀: permeability of free space = 4π * 10⁻⁷ H/m
    • N: number of turns
    • A: area (m²)
    • l: length (m)

Energy Stored

• Energy in Inductor: = ½ * L * I²
• Energy Density: = B² / (2 * μ₀)

Worked Examples

• Numerical problems involving inductance, emf, and energy calculations in solenoids and inductors.

Conclusion

• Understanding Faraday's and Lenz's laws, and their calculations is crucial for solving problems involving induced currents, transformers, and inductors.