Electromagnetic Induction
1. Introduction
Electromagnetic induction is the phenomenon of generating EMF by changing magnetic flux. It is the principle behind generators, transformers, and inductors.
2. Faraday's Laws
2.1 First Law
Whenever the magnetic flux linked with a circuit changes, an induced EMF is produced.
2.2 Second Law
The magnitude of induced EMF is equal to the rate of change of magnetic flux: ε = -dΦ/dt.
For a coil with N turns: ε = -N dΦ/dt.
2.3 Magnetic Flux
Φ = B · A = BA cos θ. SI unit: weber (Wb).
3. Lenz's Law
The induced current opposes the change in flux that produces it. This is reflected in the negative sign: ε = -dΦ/dt.
'Lenz's law is a consequence of energy conservation. The induced current always tries to maintain the original flux.'
4. Eddy Currents
When a changing magnetic flux is applied to a bulk conductor, circulating currents called eddy currents are induced in the conductor.
4.1 Effects
- Energy loss (heating) in transformers and motors — undesirable.
- Electromagnetic braking in trains — useful.
- Induction furnaces for melting metals.
4.2 Minimisation
To reduce eddy current losses, transformer cores are laminated — thin sheets insulated from each other, restricting eddy current paths.
4.3 Applications
Induction stove: Alternating current in a coil produces a changing magnetic field, inducing eddy currents in the metal pan, heating it.
Speedometer: Eddy currents in a rotating metal disc produce a torque proportional to speed.
5. Self and Mutual Inductance
4.1 Self-Inductance
L = NΦ/I. EMF induced: ε = -L dI/dt.
For a solenoid: L = μ₀n²Al, where A is area and l is length.
4.2 Mutual Inductance
M = N₂Φ₂/I₁. ε₂ = -M dI₁/dt.
For two coaxial solenoids: M = μ₀n₁n₂πr₁²l.
4.3 Energy Stored in an Inductor
U = (1/2)LI²
5. AC Generator
5.1 Principle
A coil rotating in a uniform magnetic field produces induced EMF that varies sinusoidally.
5.2 Working
As the coil rotates, the flux changes: Φ = NBA cos ωt. Induced EMF: ε = -dΦ/dt = NBAω sin ωt = ε₀ sin ωt.
5.3 Components
Field magnet, armature (coil), slip rings, brushes.
6. Worked Problems
Problem 1: A coil of 500 turns has flux changing from 0.01 Wb to 0.05 Wb in 0.1 s. Find induced EMF. Solution: ε = -N dΦ/dt = -500 × (0.05 - 0.01)/0.1 = -500 × 0.04/0.1 = -200 V. Magnitude = 200 V.
Problem 2: A solenoid of length 0.5 m, cross-section 10 cm² has 2000 turns. Find self-inductance. Solution: L = μ₀n²Al = 4π×10^{-7} × (2000/0.5)² × 10×10^{-4} × 0.5 = 4π×10^{-7} × 16×10⁶ × 5×10^{-4} = 4π×10^{-7}×8000 = 0.01005 H ≈ 10 mH.
Problem 3: An AC generator has 100 turns, area 0.1 m², rotates at 50 Hz in 0.5 T field. Find peak and RMS EMF. Solution: ω = 2πf = 100π rad/s. ε₀ = NBAω = 100×0.5×0.1×100π = 500π = 1570.8 V. ε_rms = ε₀/√2 = 1110.7 V.
7. Common Mistakes
'Students often forget that Lenz's law gives the direction of induced current. Use the right-hand rule: if flux is increasing, induced current opposes it; if decreasing, induced current aids it.'
8. ISC Exam Focus
| Topic | Theory Marks | Practical Marks |
|---|---|---|
| Faraday's and Lenz's laws | 4 | 2 |
| Self and mutual inductance | 4 | 2 |
| AC generator | 3 | 2 |
| Energy in inductor | 2 | 1 |
9. Self-Test Questions
- State and explain Faraday's laws of electromagnetic induction.
- Derive an expression for self-inductance of a long solenoid.
- Explain the working of an AC generator with a labelled diagram.
- A 50-turn coil of area 0.05 m² is placed with its plane perpendicular to a field of 0.2 T. If the field is reduced to zero in 0.1 s, find the induced EMF.
- Define mutual inductance. Derive an expression for the mutual inductance of two coaxial solenoids.
