The AC generator
A coil turning in a field
Almost all the electricity we use comes from one simple idea: turn a coil in a magnetic field. The machine is the AC generator. It changes mechanical energy into electrical energy.
Its main part is a coil, called the armature, mounted on a shaft between the poles of a magnet. The field B between the poles is uniform.
The ends of the coil are joined to two slip rings that turn with the shaft. Carbon brushes press against the rings and carry the current out to the external circuit.
The coil turns with a constant angular speed ω. Let θ be the angle between B and the normal to the coil. If θ is zero at t equals zero, then θ equals ω t.
The flux through the coil is B A cos θ, which is B A cos ω t. As the coil turns, the effective area facing the field keeps changing, so the flux keeps changing.
By Faraday’s law, for N turns, ε equals minus N dΦ_B by dt, which is N B A ω sin ω t. The largest value, ε₀, is N B A ω. So ε equals ε₀ sin ω t.
Look at the end-on view. When the plane of the coil is perpendicular to B, the flux is largest, but for a moment it is not changing. So the emf is zero.
A quarter turn later, the plane of the coil is parallel to B. The flux is zero, but it is changing fastest. Now the emf is at its peak.
After half a turn, the emf reverses. Its polarity changes back and forth with every turn, so the current alternates: alternating current, AC.
The frequency is ν equals ω over 2π, so ε equals ε₀ sin 2πνt. In India, the mains supply has a frequency of 50 hertz. In the USA it is 60 hertz.
From a bicycle to a power station
Kamla pedals a stationary bicycle. The pedals turn a coil of 100 turns and area 0.10 metre squared, in a uniform field of 0.01 tesla, perpendicular to the axis of rotation.
She pedals at half a revolution per second, so ν equals 0.5 hertz, and ω equals 2π × 0.5, which is π radians per second.
The peak emf is ε₀ equals N B A ω: 100 × 0.01 × 0.10 × 3.14, which is 0.314 volt.
What is the average emf over one complete turn? For half the turn it is positive, for the other half it is equally negative. The average over a full cycle is zero.
Now a bigger machine: 100 turns of area 0.05 metre squared in a field of 0.2 tesla, turning 50 times a second. ω equals 2π × 50, about 314 radians per second.
Then ε₀ equals 100 × 0.2 × 0.05 × 314, which is about 314 volts. More turns, a stronger field, a bigger coil, or faster turning all raise the emf.
What turns the coils in a power station? In a hydroelectric plant, falling water spins a turbine. In a thermal or nuclear plant, steam from water heated by coal or by nuclear fuel does the job. Wind turbines use the wind.
And by Lenz’s law, the current in the coil opposes the rotation. The more current drawn, the harder the turbine must be driven. The electrical energy we use is paid for with mechanical work.
Try it: turn the generator
- This is Kamla’s bicycle generator: peak emf ε₀ = NBAω = 0.314 V.
- Press +45° step by step. At 0° and 180° the coil faces the field: flux is largest and the emf is zero. At 90° and 270° the flux is zero and the emf is largest.
- The dashed curve (flux linkage × ω) is a quarter cycle ahead of the emf: ε is the negative slope of NΦ_B.
- Double the frequency. The peak emf doubles and the period halves.
Kamla’s bicycle
Kamla pedals at 0.5 revolution per second, turning a coil of 100 turns and area 0.10 m² in a field of 0.01 T perpendicular to the axis of rotation. What is the maximum emf?
- 0.05 V
- 0.157 V
- 0.314 V
- 0.628 V
Answer
C. ω = 2πν = 2π × 0.5 = π rad/s. ε₀ = NBAω = 100 × 0.01 × 0.10 × 3.14 = 0.314 V. Using ω = 0.5 (forgetting the 2π) gives 0.05 V; taking ν = 1 Hz gives 0.628 V.
Coil facing the field
At the instant when the plane of a generator coil is perpendicular to the magnetic field, which is true?
- The flux is zero and the emf is maximum
- The flux is maximum and the emf is zero
- Both the flux and the emf are maximum
- Both the flux and the emf are zero
Answer
B. With the plane ⟂ B, the normal is along B (θ = 0°), so Φ_B = BA cos 0° is maximum. But cos θ is flat at its peak: the flux is momentarily not changing, so ε = ε₀ sin 0° = 0. The emf is largest a quarter turn later, when the flux is zero but changing fastest.
Summary
- Flux through the coil:
Φ_B = BA cos ωt - Induced emf:
ε = NBAω sin ωt = ε₀ sin ωt - Peak emf:
ε₀ = NBAω - Frequency:
ω = 2πν
- An AC generator turns a coil (armature) in a uniform field; slip rings and brushes connect it to the outside circuit.
- The emf is zero when the coil’s plane is ⟂ B (flux maximum) and maximum when its plane is ∥ B (flux zero).
- The emf reverses every half turn, giving alternating current; its average over a full cycle is zero. Indian mains: 50 Hz.
- The mechanical energy comes from turbines driven by falling water, steam or wind; by Lenz’s law, drawing current makes the coil harder to turn.