Learning Objectives
- Understand magnetic flux and Faraday's laws of electromagnetic induction
- Apply Lenz's law to determine direction of induced EMF
- Study motional EMF and eddy currents
- Understand self-inductance and mutual inductance
- Learn about the energy stored in an inductor
Key Concepts
Magnetic Flux
Φ = B · A = BA cos θ (SI unit: weber, Wb = T·m²)
Flux is maximum when field is perpendicular to the surface area (θ = 0°) and zero when parallel (θ = 90°).
Faraday's Laws
First Law: Whenever the magnetic flux through a circuit changes, an EMF is induced.
Second Law: The magnitude of induced EMF equals the rate of change of magnetic flux.
ε = -dΦ/dt (for a single loop)
ε = -N dΦ/dt (for N turns)
Lenz's Law
The direction of induced current is such that it opposes the change in magnetic flux that caused it. This is a consequence of conservation of energy.
The negative sign in Faraday's law represents Lenz's law.
Motional EMF
EMF induced in a conductor moving in a magnetic field:
ε = BLv (for a rod of length L moving with velocity v perpendicular to B).
ε = BLv sin θ (when velocity makes angle θ with the rod).
For a rotating coil: ε = NBAω sin ωt = ε₀ sin ωt (AC generator principle).
Eddy Currents
Circulating currents induced in bulk conductors when exposed to changing magnetic flux. They cause heating (energy loss).
Applications: Electromagnetic braking, induction furnace, electric power meters, induction cooktops.
Minimisation: Using laminated cores (thin insulated sheets).
Self-Inductance
Property of a coil by which it opposes change in current through itself by inducing an EMF.
Φ = LI → ε = -L dI/dt
SI unit: henry (H). L depends on geometry, number of turns, and core material.
Solenoid: L = μ₀n²Al = μ₀N²A/l (n = N/l = turns per unit length).
Mutual Inductance
EMF induced in one coil due to change of current in another nearby coil.
Φ₂ = MI₁ → ε₂ = -M dI₁/dt
Two co-axial solenoids: M = μ₀n₁n₂Al (A is area of inner solenoid).
M₁₂ = M₂₁ = M. For coupled coils: M = k√(L₁L₂), where k is coupling coefficient (0 ≤ k ≤ 1).
Energy Stored in an Inductor
U = ½LI²
Energy density: u = B²/(2μ₀) (energy per unit volume in magnetic field).
Summary
Electromagnetic induction is the process of generating EMF by changing magnetic flux. Faraday's law gives the magnitude and Lenz's law gives the direction. Motional EMF arises when conductors move in magnetic fields (principle of generators). Self-inductance opposes current changes in a coil. Mutual inductance couples two coils. Energy stored in an inductor is ½LI².
Important Terms
- Magnetic Flux: Φ = BA cos θ, measured in weber (Wb)
- Faraday's Law: ε = -NdΦ/dt
- Lenz's Law: Induced current opposes the cause
- Self-Inductance: Coil's opposition to current change, ε = -LdI/dt
- Mutual Inductance: Coupling between two coils, ε = -MdI/dt
- Eddy Currents: Circulating currents in bulk conductors
Quick Revision
- Φ = BA cos θ; ε = -NdΦ/dt
- Motional EMF: ε = BLv; Generator: ε = NBAω sin ωt
- Self-inductance: L = μ₀n²Al; ε = -LdI/dt
- Mutual inductance: M = μ₀n₁n₂Al; M = k√(L₁L₂)
- Energy: U = ½LI²; Energy density: u = B²/2μ₀
- Lenz's law ensures conservation of energy