Class 12 · Chapter 6

Electromagnetic Induction

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Electromagnetic Induction Short Notes

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Condensed revision points for Electromagnetic Induction — for quick recall before exams, not a substitute for the full notes.

Φ = B·A = BAcosθ (θ from the normal); SI unit weber (Wb), 1 Wb = 1 T·m². For N turns, flux LINKAGE is NΦ.
Flux through any CLOSED surface is always zero (Gauss's law for magnetism) — field lines never begin or end.
Faraday's law: ε = −dΦ/dt (single turn), ε = −N(dΦ/dt) (N turns). Depends only on the RATE of change of flux, never its magnitude.
Doesn't matter WHY flux changes — moving magnet, changing current nearby, rotating coil, or changing area/shape — the law applies identically.
An open circuit still develops an induced EMF (measurable across the break); it just can't sustain a current.

Induced current opposes the CHANGE in flux (not the field itself) — a restatement of energy conservation, built into Faraday's law as the negative sign.
Flux increasing ⟹ induced current opposes the increase. Flux decreasing ⟹ induced current tries to maintain it.
Pulling a magnet toward a coil always meets repulsion; pushing it away always meets attraction — mechanical work done = electrical energy generated.
Fastest way to get the DIRECTION of induced current in conceptual questions — no calculation needed.

Rod on rails (v, B, l mutually ⊥): ε = Bvl, I = Bvl/R, retarding force F = BIl = B²l²v/R.
Two equivalent derivations: (1) Faraday's law — area swept changes at rate lv, so dΦ/dt = Blv. (2) Force on free charges — F/q = v×B pushes charges to one end until equilibrium; ε = ∫(v×B)·dl = Bvl.
Rod rotating about one end: ε = ½Bωl² — each element at radius r moves at ωr, contributing dε = B(ωr)dr, integrated 0 to l. Direct preview of the AC generator.
Constant-velocity rod: applied force must exactly balance B²l²v/R at every instant — this IS the simplest working generator.

Swirling induced currents inside bulk conductors (not confined to a thin wire) — always oppose the motion/change causing them (Lenz's law).
Useful: induction furnaces, induction cooktops, magnetic braking (trains, roller coasters), older speedometers.
Unwanted: wasted heating in transformer cores — minimised with a LAMINATED (thin, insulated sheets) core instead of a solid block.
Not a separate phenomenon — just ordinary induced current flowing in loops through bulk material rather than a designed wire circuit.

Φ = LI; self-induced EMF ε = −L(dI/dt). L depends only on geometry & core material — never on I itself (like capacitance C).
Opposes the CHANGE in current, not the current — no effect on steady current, resists sudden changes ('electrical inertia').
SI unit henry (H); 1 H = 1 Wb/A = 1 V·s/A.
Long solenoid: L = μ₀n²Al = μ₀N²A/l. Derivation: B=μ₀nI inside, flux linkage NΦ=(nl)(BA)=μ₀n²AlI. Ferromagnetic core multiplies L by μᵣ.
L ∝ n² — doubling turn density quadruples L for the same solenoid length and area.

Φ₂ = MI₁; mutually induced EMF ε₂ = −M(dI₁/dt). M depends on both coils' geometry AND relative position/orientation.
Reciprocity: M₁₂ = M₂₁ always — one single M describes the coupling regardless of which coil drives the other.
Entire operating principle of a TRANSFORMER — AC in the primary induces AC EMF in the secondary purely through mutual coupling, no direct electrical connection.
Two coaxial solenoids: M = μ₀N₁N₂A/l — same pattern as self-inductance: find the field from one coil, then the flux it produces through the other's turns.

No mutual coupling: series L(eq) = L₁+L₂+⋯ (like resistors); parallel 1/L(eq) = 1/L₁+1/L₂+⋯. Opposite pattern to capacitors.
Energy stored in an inductor: U = ½LI² (derived from P=εI=L(dI/dt)·I, integrated 0 to I). Structurally identical to a capacitor's U=½CV².
Energy density of magnetic field: u = B²/2μ₀ — directly parallel to u=½ε₀E² for the electric field.
Breaking a current-carrying inductive circuit suddenly can spike voltage dangerously — stored energy must dissipate quickly as di/dt spikes.

Growth (battery connected): I(t) = I₀(1−e^(−t/τ)), τ = L/R, I₀ = ε/R. From ε = IR + L(dI/dt).
At t=τ, current reaches 63.2% of final value; never reaches I₀ exactly, only asymptotically. Same shape as RC charging.
Decay (short-circuited): I(t) = I₀e^(−t/τ). At t=τ, current falls to 36.8% of initial value — mirror of growth, same shape as RC discharge.
Larger L or smaller R ⟹ larger τ ⟹ slower current buildup/decay. Heat dissipated during full decay = ½LI₀² exactly (energy conservation).

Converts mechanical → electrical energy by rotating a coil in a field (or magnet inside a fixed coil). ε = ε₀sin(ωt), peak ε₀ = NBAω.
Components: field magnet, armature (rotating coil), slip rings, carbon brushes.
Derivation: Φ=BAcos(ωt) ⟹ ε=−N(dΦ/dt)=NBAωsin(ωt).
EMF is MAX when coil plane is PARALLEL to B (flux changing fastest); EMF is ZERO when coil plane is PERPENDICULAR to B (flux itself at its own max, but momentarily not changing) — a common trap.
Grid frequency fixed by mechanical rotation speed — standardised at 50 Hz in India.

Generators/alternators at power stations (any turbine source — water, steam, wind).
Transformers: mutual inductance steps voltage up/down for efficient transmission and safe household use.
Induction cooktops/furnaces: eddy currents heat the vessel/workpiece directly, no flame or resistive element.
Metal detectors, magnetic braking, wireless (inductive) charging — all rely on changing magnetic flux inducing usable current or force.

Lenz's law opposes the CHANGE in flux, not the field itself — the induced field can point the SAME way as B if flux is decreasing.
EMF peaks where flux is changing FASTEST (coil plane ∥ B), not where flux itself is maximum (coil plane ⊥ B) — very commonly reversed by mistake.
Self-induced EMF opposes change in I, not I itself — a steady current through an inductor produces zero self-induced EMF.
Inductors combine like resistors (series adds, parallel adds reciprocals) — capacitors do the OPPOSITE. Easy to swap under exam pressure.
τ=L/R for LR circuits is the direct analogue of τ=RC for RC circuits, but note L/R (not L×R) — a common substitution slip.
An open induced circuit still has an EMF across the break, just no current — flux still changing even if nothing flows.

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