What this chapter is about
Electromagnetic induction is the phenomenon where a changing magnetic flux through a conducting loop produces an electromotive force (emf) in that loop. This chapter builds on your understanding of magnetic fields from earlier work and introduces how electricity can be generated from magnetism. The discovery by Michael Faraday in 1831 revolutionised technology and led to electric generators, transformers and countless devices we use today.
A Class 12 student meets this chapter now because it connects electricity and magnetism into a unified picture. You have studied how currents produce magnetic fields; here you learn the reverse — how changing magnetic fields produce currents. After studying this chapter, you should be able to state Faraday's law and Lenz's law precisely, calculate induced emf in various situations, understand self-inductance and mutual inductance, and explain how energy is stored in a magnetic field.
This chapter also prepares you for alternating current circuits, where induced emf varies sinusoidally with time. The mathematical tools involve differentiation (rate of change of flux) and integration (energy stored), connecting physics to calculus in a practical way.
Key ideas
- Magnetic flux through a surface equals the product of the magnetic field component perpendicular to the surface and the area: Φ = B A cos θ, measured in weber (Wb).
- Faraday's law of electromagnetic induction states that the magnitude of induced emf in a loop equals the rate of change of magnetic flux through the loop: |ε| = |dΦ/dt|.
- Lenz's law gives the direction of induced emf: the induced current opposes the change in flux that causes it, ensuring energy conservation.
- Motional emf arises when a conductor moves through a magnetic field; for a rod of length l moving with velocity v perpendicular to field B, the emf is ε = Blv.
- Eddy currents are circulating currents induced in bulk conductors when flux through them changes; they cause heating and are used in induction furnaces and electromagnetic braking.
- Self-inductance (L) of a coil measures how the coil opposes change in its own current; the induced emf is ε = −L (dI/dt), and the SI unit is henry (H).
- Mutual inductance (M) describes flux linkage between two coils; an emf is induced in one coil when current in the neighbouring coil changes.
- Energy stored in an inductor carrying current I is U = (1/2) L I², analogous to energy stored in a capacitor.
Formulas and facts to remember
- Magnetic flux: Φ = B A cos θ, where θ is the angle between B and the area normal.
- Faraday's law: ε = −dΦ/dt (the negative sign represents Lenz's law).
- For N turns: ε = −N (dΦ/dt).
- Motional emf for a rod: ε = Blv (when B, l and v are mutually perpendicular).
- Self-inductance of a solenoid: L = μ₀ n² A l, where n is turns per unit length, A is cross-sectional area and l is length.
- Mutual inductance: ε₂ = −M (dI₁/dt).
- Energy stored in an inductor: U = (1/2) L I².
- 1 weber = 1 volt × 1 second; 1 henry = 1 weber per ampere.
Worked examples
Example 1: Calculating induced emf from changing flux
A circular coil of 50 turns and radius 0.10 m is placed perpendicular to a uniform magnetic field. The field increases uniformly from 0.20 T to 0.50 T in 0.05 s. Find the induced emf.
Solution: Area A = π r² = π × (0.10)² = 0.0314 m². Initial flux Φ₁ = B₁ A = 0.20 × 0.0314 = 0.00628 Wb. Final flux Φ₂ = 0.50 × 0.0314 = 0.0157 Wb. Change in flux ΔΦ = 0.0157 − 0.00628 = 0.00942 Wb. Rate of change dΦ/dt = 0.00942 / 0.05 = 0.188 Wb/s. Induced emf = N × (dΦ/dt) = 50 × 0.188 = 9.4 V.
Example 2: Motional emf in a sliding rod
A metal rod 0.40 m long slides on two parallel rails at 5.0 m/s in a region where the magnetic field is 0.30 T, directed perpendicular to the plane of the rails. Calculate the emf induced across the rod.
Solution: Using ε = Blv: ε = 0.30 × 0.40 × 5.0 = 0.60 V. The rod acts like a seat of emf; current flows if the circuit is closed through the rails.
Example 3: Energy stored in an inductor
A coil has self-inductance 2.0 H and carries a steady current of 3.0 A. How much energy is stored in its magnetic field?
Solution: U = (1/2) L I² = 0.5 × 2.0 × (3.0)² = 0.5 × 2.0 × 9.0 = 9.0 J.
Common mistakes
- Forgetting the negative sign in Faraday's law → the sign encodes Lenz's law and shows opposition to flux change; always interpret direction using it.
- Using total magnetic field instead of the perpendicular component → include cos θ when the field is not normal to the surface.
- Confusing flux (Φ = BA cos θ) with field (B) → flux depends on area and orientation, not just field strength.
- Ignoring the number of turns when applying Faraday's law → multiply dΦ/dt by N for a coil with N turns.
- Thinking induced current appears even when the circuit is open → an emf is induced, but current flows only if a closed path exists.
Quick revision
- Induced emf equals rate of change of magnetic flux: ε = −N (dΦ/dt).
- Lenz's law: induced effects always oppose the cause producing them.
- Motional emf for a rod: ε = Blv when B, l and v are mutually perpendicular.
- Self-inductance opposes change in current; unit is henry (H).
- Energy in an inductor: U = (1/2) L I².
- Eddy currents arise in bulk conductors and cause heating; laminated cores reduce them.