JEE/NEET Physics · Electromagnetic Induction series · Part 6 of 6 · All parts →
- Generator = coil rotating in field: flux oscillates sinusoidally → AC voltage
- ε = NBAω sin(ωt): peak ε₀ = NBAω
- Frequency of rotation = frequency of the AC (50 Hz needs 50 revs/s)
- Converts mechanical → electrical via the grid’s turbines
- The whole series on one card — flux, Lenz, motional EMF, eddies, inductance
Spin a coil in a magnetic field and its flux swings like a sine wave — out comes alternating voltage. Every power plant since Faraday is this machine scaled up: the turbine-generator feeding whatever device you’re reading this on. Part 6 of the Electromagnetic Induction series.
- The rotating coil
- The sine-wave EMF
- From water wheel to wall socket
- The master formula card
- Final practice set
- Recap
The Rotating Coil
Flux through a rotating coil: Φ = NBA cos(ωt) — oscillating as the coil’s face swings through the field. Differentiate (Faraday) and out comes:
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| ω | rotation rate (angular) | rad/s; f = ω/2π Hz |
| ε₀ | peak (maximum) EMF | V |
| N, B, A | turns, field, coil area | — |
From Water Wheel to Wall Socket
Steam or water spins a turbine; the turbine spins coils (or field magnets); ε₀ sin(ωt) flows to the grid; transformers (mutual inductance) step voltage up for travel and down for use. One rotating machine and one coupled pair run the electrified world.
The Master Formula Card
| What | Formula | Remember |
|---|---|---|
| Flux | Φ = BA cosθ | weber |
| Faraday | ε = −N dΦ/dt | change is all |
| Lenz | opposes the change | energy honesty |
| Motional EMF | Bℓv | ⊥ trio |
| Rail retarding force | B²ℓ²v/R | generator hides motor |
| Rotating rod | ½Bωℓ² | swept area |
| Eddies | bulk swirls, Lenz-opposed | laminate cores |
| Self-inductance | ε = −L dI/dt | electrical inertia |
| Solenoid L | μ₀n²Al | n² leverage |
| Stored energy | ½LI² | field battery |
| Mutual | ε₂ = −M dI₁/dt | transformer talk |
| Generator | ε = NBAω sinωt | ε₀ = NBAω |
Solved Examples
ε₀ = NBAω = 100 × 0.3 × 0.01 × 50 = 150 V.
✔
Answer: 150 V
ε₀ doubles to 300 V; f = ω/2π rises from ~8 Hz to ~16 Hz.
Speed buys both voltage and frequency.
✔
Answer: 300 V at 16 Hz
50 revolutions/s × 60 = 3000 RPM: rotation frequency must equal grid frequency.
The whole plant’s engineering — turbine blades to boiler pressure — is locked to this one number.
✔
Answer: 50 Hz = 3000 RPM (2-pole)
- Using cos for the EMF. Flux goes as cos(ωt); the EMF (its derivative) as sin: the 90° shift is standard exam bait.
- Peak vs RMS confusion. ε₀ is the peak; the next series (AC) defines the effective values — don’t mix them here.
- Doubling ω ‘doubles energy’. It doubles peak voltage AND frequency: power quadruples per cycle count — read the question’s wording carefully.
- Forgetting the retarding torque. A loaded generator resists its turbine (the hidden motor): power plants burn fuel fighting their own output — by design.
This Physics in Your Daily Life
- Every wall socket — NBAω sin(ωt) at 230 V, 50 Hz, from coal, gas, nuclear, hydro, wind, or sun: one machine, many masters.
- Bicycle dynamos and emergency hand-crank radios — your muscles as the turbine: Faraday’s machine shrunk to pocket size.
- Wind turbines — giant rotating flux-changers: climate policy riding on a 19th-century coil.
- Backup generators in hospitals — diesel turbines spinning coils within seconds of a mains cut: lives bridged by NBAω.
- The entire AC-vs-DC history (Tesla/Edison) — induction machines favour AC (transformers need change): the war of currents was really a referendum on Faraday’s law.
A water wheel lifts buckets up, over, down, and around: the height of any one bucket traces a smooth repeating wave. A coil’s flux does the same — swinging from maximum (face-on) to zero (edge-on) to maximum-reversed each half turn: differentiate the circle’s projection and you differentiate cos into sin: AC is geometry in motion.
100 turns, 0.01 m², 0.3 T at 50 rad/s: peaks of 150 V swinging positive-negative 8 times a second. Double the spin: 300 V peaks, 16 swings — voltage and frequency locked to RPM: why grid turbines are speed-disciplined to 0.1%.
Plot flux (cos) and EMF (sin) on the same time axis: when flux is maximum, its slope — the EMF — is zero; when flux sweeps fastest through zero, EMF peaks. The two curves’ offset dance is the calculus of induction, made visible.
Practice set (answers hidden — try first)
(NEET-level) ε₀ for N=50, B=0.4, A=0.01 m², ω=100:
(JEE Main-level) Doubling ω does what to ε₀ and f?
(NEET-level) EMF is max when flux is
(Concept) Transformers work on AC because induction needs
(JEE Main-level) 2-pole turbine for 50 Hz: RPM =
- Φ = NBAcosωt → ε = NBAω sinωt
- ε₀ = NBAω
- rotation frequency = AC frequency
- loaded generators resist turbines
- flux and EMF 90° out of phase
- 🔁 generator EMF formula
- 🔁 flux-EMF phase relation
- 🔁 grid-turbine speed link
- 🧠 Chant: ‘N-B-A-omega: the peak recipe’.
- 🧠 Grid lock: ’50 Hz = 3000 RPM (2-pole)’.
- 🏠 Daily: every socket serves NBAω sinωt.
- 🏠 Daily: transformers need change — that’s why AC won.
Quick revision
- Generator = coil rotating in field: flux oscillates sinusoidally → AC voltage
- ε = NBAω sin(ωt): peak ε₀ = NBAω
- Frequency of rotation = frequency of the AC (50 Hz needs 50 revs/s)
- Converts mechanical → electrical via the grid’s turbines
- The whole series on one card — flux, Lenz, motional EMF, eddies, inductance
- From water wheel to wall socket
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