Electromagnetic Induction
'If you can change a magnetic field, you can create electricity. This is INDUCTION — and it powers the modern world.'
1. Chapter Overview
Electromagnetic induction is the phenomenon of producing an INDUCED EMF (and current) when the magnetic flux through a circuit CHANGES. Topics include: FARADAY'S LAWS of electromagnetic induction, LENZ'S LAW (determining the direction of induced current), SELF-INDUCTANCE (the property of a coil to oppose changes in its own current), MUTUAL INDUCTANCE, the AC GENERATOR, and EDDY CURRENTS.
2. Magnetic Flux
- Φ_B = B · A = BA cos θ. Unit: Weber (Wb).
- 'Flux is the "amount" of magnetic field PASSING THROUGH a surface.'
- Φ_B = NBA cos θ for an N-turn coil.
3. Faraday's Laws
First Law
- 'Whenever the magnetic flux linked with a circuit CHANGES, an induced EMF is produced.'
Second Law
- ε = −dΦ/dt (for a single turn). ε = −N dΦ/dt (for an N-turn coil).
- 'The magnitude of the induced EMF equals the RATE OF CHANGE of magnetic flux.'
Methods of Changing Flux
- Change the magnetic field B.
- Change the area A of the loop.
- Change the angle θ between B and the normal to the loop.
4. Lenz's Law
- 'The direction of induced current is such that it OPPOSES the CAUSE that produces it.'
- ε = −dΦ/dt — the NEGATIVE SIGN IS LENZ'S LAW.
- 'Lenz's law is ENERGY CONSERVATION in action. If induced current HELPED the change, we would get energy from nothing.'
Worked Example 1
Problem: A coil of 100 turns has a flux of 0.05 Wb through it. If the flux is reduced to zero in 0.1 s, find the induced EMF. Solution: ε = −N dΦ/dt = −100(0 − 0.05)/0.1 = −100(−0.5) = 50 V. 'The magnitude is 50 V.'
5. Self-Inductance
- L = NΦ/I. 'The property of a coil to oppose changes in its own current.'
- Induced EMF: ε = −L dI/dt.
- Inductance of a solenoid: L = μ₀N²A/l.
- Unit: Henry (H). 1 H = 1 Wb/A.
Energy Stored in an Inductor
- U = ½ LI². 'An inductor stores energy in its MAGNETIC FIELD.'
6. Mutual Inductance
- M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂.
- 'The flux through coil 2 due to current in coil 1 — divided by that current.'
- Induced EMF: ε₂ = −M dI₁/dt.
- M for two coaxial solenoids: M = μ₀N₁N₂A/l (if inner solenoid is inside outer one).
7. AC Generator
- Principle: Electromagnetic induction — rotating a coil in a magnetic field.
- EMF: ε = NBAω sin(ωt) = ε₀ sin(ωt). 'The EMF varies SINUSOIDALLY with time.'
- Peak EMF: ε₀ = NBAω. RMS EMF: ε_rms = ε₀/√2.
| Component | Function |
|---|---|
| Field magnet | Provides uniform magnetic field (typically N and S poles) |
| Armature (coil) | Rotates in the field — where EMF is induced |
| Slip rings | Connect rotating coil to external circuit — maintain contact |
| Brushes | Stationary contacts that press against slip rings |
8. Eddy Currents
- 'When a conducting material moves through a changing magnetic field, INDUCED CURRENTS flow within the material itself — like whirlpools in water.'
- Effects: Heating (induction furnace), braking (eddy current brakes in trains), damping (galvanometer damping).
- Minimising: Laminate the conductor (thin sheets insulated from each other) — breaks the path of eddy currents.
9. Comparison Table: Self-Inductance vs Mutual Inductance
| Feature | Self-Inductance (L) | Mutual Inductance (M) |
|---|---|---|
| Definition | Opposition to change in OWN current | EMF induced in ANOTHER coil due to change in THIS coil |
| Symbol | L | M |
| Formula | L = NΦ/I | M = N₂Φ₂₁/I₁ |
| Depends on | Own geometry and core | Geometry of BOTH coils, their separation, and core |
| Unit | Henry (H) | Henry (H) |
10. Common Mistakes
- The negative sign in Faraday's law: It is NOT optional. It embodies LENZ'S LAW — the induced EMF opposes the change.
- Induced EMF vs induced current: Induced EMF depends ONLY on rate of change of flux. Induced current = EMF / resistance.
- Self-inductance formula: L = μ₀N²A/l — number of turns is SQUARED, not linear.
- AC generator EMF: Peak value = NBAω. RMS value = NBAω/√2. Many students forget the relationship.
11. CBSE Exam Focus
- Faraday's laws — magnitude of induced EMF (ε = −N dΦ/dt)
- Lenz's law — direction of induced current
- Self-inductance — L = μ₀N²A/l for solenoid, energy stored (½LI²)
- Mutual inductance — M for coaxial solenoids
- AC generator — construction, working, EMF = NBAω sin(ωt)
- Eddy currents — effects, applications, minimisation
12. Self-Test
Q1: A rectangular loop of area 0.1 m² is placed perpendicular to a uniform magnetic field of 0.5 T. The field is reduced to zero in 0.01 s. Find the induced EMF. A1: ε = −dΦ/dt = −(0 − 0.5×0.1)/0.01 = −(−0.05)/0.01 = 5 V.
Q2: A solenoid of length 0.5 m, area 4×10⁻⁴ m², and 1000 turns carries 2 A. Find its self-inductance. A2: L = μ₀N²A/l = (4π×10⁻⁷)(1000)²(4×10⁻⁴)/0.5 = (4π×10⁻⁷×10⁶×4×10⁻⁴)/0.5 = (4π×4×10⁻⁵)/0.5 = 16π×10⁻⁵/0.5 = 32π×10⁻⁵ = 1.005×10⁻³ H = 1 mH.
Q3: A 10 mH inductor has a current that changes from 5 A to 1 A in 0.02 s. Find the induced EMF. A3: ε = −L dI/dt = −10×10⁻³(1−5)/0.02 = −10⁻²(−4)/0.02 = 0.04/0.02 = 2 V.
Q4: An AC generator has 200 turns, area 0.05 m², in a 0.2 T field, rotating at 50 rad/s. Find peak and RMS EMF. A4: ε₀ = NBAω = 200×0.2×0.05×50 = 100 V. ε_rms = ε₀/√2 = 100/√2 = 70.7 V.
Q5: Two coils have mutual inductance 0.05 H. If the current in the primary changes from 4 A to 1 A in 0.01 s, find the induced EMF in the secondary. A5: ε₂ = −M dI₁/dt = −0.05(1−4)/0.01 = −0.05(−3)/0.01 = 0.15/0.01 = 15 V.
13. Conclusion
Electromagnetic induction is how we GENERATE electricity:
- FARADAY: 'Change the flux — get an EMF. The RATE of change determines the VOLTAGE.'
- LENZ: 'Nature OPPOSES change — the induced current tries to maintain the status quo.'
- INDUCTANCE: 'A coil resists changes in current — the property of SELF-INDUCTANCE is electrical inertia.'
- GENERATOR: 'Mechanical energy → Electrical energy. The reverse of a motor.'
'Electromagnetic induction is the PRINCIPLE behind the electric generator — and the reason our civilisation runs on electricity.'
