AC generator · the dynamo

A loop of wire spins steadily in a fixed magnetic field — the world's recipe for electricity. The voltage it makes alternates. Spin it faster and you obviously get more cycles per second — but does the PEAK voltage rise too, or is it fixed by the field and the loop?

▶ Run the simulationSee the measured result

Units: V — the peak EMF ε₀ = N B A ω = 50π V of the real bench (100 turns · 0.25 T · 0.02 m² · 2π·50 Hz), the textbook dynamo prediction (Griffiths §7.1.3)

How the lab tests it

Rotate a loop (N turns, area A) at angular velocity ω in a uniform field B, so the flux linkage Φ(t) = N B A cos ωt sweeps a full cosine each turn. The lab traces Φ(t) against the induced EMF = −dΦ/dt, marks the loop's face-on (Φ extremum) and edge-on (Φ = 0) instants, and overlays the SAME loop spun at 2ω. A real generator — N=100, A=0.02 m², B=0.25 T, f=50 Hz — is also read with 2% noise on its peak.

What it checks

EMF = N B A ω sin ωt — a SINE 90° behind the cosine flux, so the voltage is exactly ZERO when the loop is face-on (flux MAXIMUM, conductors sliding along the field) and PEAKS when the loop is edge-on (flux zero, conductors cutting fastest): the rotating, steady-state form of 'rate, not flux'. And the peak ε₀ = N B A ω is PROPORTIONAL to ω — double the spin, double the frequency AND double the amplitude (why a bicycle dynamo brightens, not just flickers); inverting a real ε₀ ≈ 157 V weighs the field B = ε₀/(N A·2πf) ≈ 0.25 T

AC generator EMF calculator

What a spinning coil puts out. The peak voltage of an AC generator is fixed by four things you can measure with a ruler, a gaussmeter and a tachometer — turns, field, area and crank rate — and the simulation above recovers exactly that from nothing but the Lorentz force (v×B)·dl on four straight wires: no sinusoid, no flux formula and no NBAω anywhere in its recovery path, which is what makes the agreement worth anything. Neither number this calculator turns on is typed in here either: ω is assembled as 2πf and the rms factor as ε₀/√2, never as 6.2832 or 0.7071. The defaults are the lab's own bench — 100 turns, 0.25 T, 0.02 m², 50 Hz → 157.08 V peak, 111 V rms, a mains-style line — and the default "measured" 160.029 V is what the simulation's own noisy meter actually reads, so inverting it returns B̂ = 0.254694 T, the +1.88% meter draw that world discloses on screen rather than hides. Four things this lab does NOT measure, so the calculator does not pretend to: the load — this is the open-circuit EMF, with no armature resistance, no current, and no Lenz braking torque telling you what the machine costs to turn; iron and the shaped pole faces that make a real alternator's gap field non-uniform, so a real waveform carries harmonics a uniform field cannot produce; the winding's own inductance and the power factor it creates under load; and eddy-current and hysteresis losses. What it does measure, and what this relation therefore rests on, is in the finding: the sum is a pure sinusoid at exactly the crank rate, 90° behind the flux, with ε₀ strictly proportional to ω, B and A.