Unit 13: Electromagnetic Induction

1. Faraday's Law of Induction

Faraday's Law describes how changing magnetic fields induce electromotive force (EMF) in conductors.

ε = -N(dΦ/dt)
where:
ε = induced EMF
N = number of turns in coil
Φ = magnetic flux
dΦ/dt = rate of change of magnetic flux
Methods of Inducing EMF:
  • Changing magnetic field strength
  • Changing loop area
  • Changing orientation (angle)
  • Relative motion between field and conductor

2. Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a surface area.

Φ = BA cos θ
where:
B = magnetic field strength
A = area of loop/surface
θ = angle between B and surface normal
Example: A 20 cm² loop is oriented 30° to a 0.5T magnetic field. Calculate:
a) Magnetic flux
b) Change in flux if field increases to 0.8T in 0.1s

3. Lenz's Law

Lenz's Law determines the direction of induced current by stating that it opposes the change causing it.

Applying Lenz's Law:
  • Identify change in magnetic flux
  • Determine direction that opposes change
  • Use right-hand rule for current direction
  • Conservation of energy is maintained
Lenz's Law demonstration

4. Motional EMF

Motional EMF is induced when a conductor moves through a magnetic field.

ε = Bℓv
where:
B = magnetic field
ℓ = length of conductor
v = velocity perpendicular to field
Example: A 50cm conductor moves at 10m/s perpendicular to a 0.2T magnetic field. Calculate:
a) Induced EMF
b) Current in a 5Ω circuit

5. Self-Inductance

Self-inductance is the property of a circuit to oppose changes in current through electromagnetic induction.

V = L(dI/dt)
L = N²μ₀A/ℓ (solenoid)
Energy stored = ½LI²
Properties of Inductors:
  • Opposes changes in current
  • Stores energy in magnetic field
  • Acts like open circuit initially
  • Measured in henries (H)

6. Mutual Inductance

Mutual inductance occurs when current changes in one circuit induce EMF in another nearby circuit.

ε₂ = -M(dI₁/dt)
M = k√(L₁L₂)
where:
M = mutual inductance
k = coupling coefficient (0 ≤ k ≤ 1)

7. RL Circuits

RL circuits exhibit time-dependent behavior due to inductance.

I(t) = (V₀/R)(1 - e^(-Rt/L)) (charging)
I(t) = I₀e^(-Rt/L) (discharging)
Time constant: τ = L/R
Example: In an RL circuit with L = 0.5H and R = 100Ω:
a) Find time constant
b) Time to reach 63.2% of final current

8. Energy Transfer and Conservation

Understanding energy transfer in electromagnetic induction is crucial for practical applications.

Energy Considerations:
  • Mechanical energy → Electrical energy
  • Electrical energy → Magnetic field energy
  • Conservation of energy applies
  • Power transfer in transformers

9. Applications

Electromagnetic induction has numerous practical applications in modern technology.

Common Applications:
  • Electric generators
  • Transformers
  • Induction cooktops
  • Wireless charging
  • Metal detectors
  • Guitar pickups

10. Practice Problems

Problem 1: A square loop of side 10cm rotates at 60 rpm in a 0.5T magnetic field. Find:
a) Maximum flux
b) EMF as function of time
c) Maximum EMF induced
Problem 2: A solenoid has 500 turns and inductance 0.8H. Calculate:
a) EMF induced by current changing at 100A/s
b) Energy stored at 2A
c) Current after one time constant if R = 40Ω
Problem-Solving Strategy:
  • Draw clear diagrams
  • Identify changing quantities
  • Apply Faraday's and Lenz's laws
  • Check units and signs
  • Verify reasonable answers